Lesson Explainer: Graphs of Logarithmic Functions Mathematics

In this explainer, we will learn how to sketch logarithmic functions with different bases and their transformations, and we will study their different characteristics.

Letโ€™s start by recalling the definition of a logarithmic function.

Definition: Logarithmic Function

A logarithmic function is the inverse of an exponential function. For the exponential function ๐‘“(๐‘ฅ)=๐‘›๏—, where ๐‘›>0 and ๐‘›โ‰ 1, the inverse logarithmic function is ๐‘“(๐‘ฅ)=๐‘ฅ๏Šฑ๏Šง๏Šlog.

If the point (๐‘ฅ,๐‘ฆ) satisfies the exponential function, then the point (๐‘ฆ,๐‘ฅ) satisfies the logarithmic function. That is, if ๐‘ฆ=๐‘›๏—, then ๐‘ฅ=๐‘ฆlog๏Š.

Since a logarithmic function is the inverse function of an exponential function, and the graphs of inverse functions are reflections in the line ๐‘ฆ=๐‘ฅ, we can sketch a graph of ๐‘ฆ=๐‘ฅlog๏Š by reflecting an exponential curve.

Letโ€™s sketch the graph of ๐‘ฆ=๐‘ฅlog๏Šง๏Šฆ, which we can also write as ๐‘ฆ=๐‘ฅlog. To do this, we will start by sketching the graph of ๐‘ฆ=10๏—.

We can see that the exponential function ๐‘“(๐‘ฅ)=10๏— has a domain of ]โˆ’โˆž,โˆž[ and a range of ]0,โˆž[. Its graph has the negative ๐‘ฅ-axis as an asymptote, and no portion of the graph lies below the ๐‘ฅ-axis. This is because 10 raised to a power can never be negative or 0. The graph has a ๐‘ฆ-intercept of 1, since ๐‘“(0)=10=1๏Šฆ, and it passes through the point (1,10), since ๐‘“(1)=10=10๏Šง. Also, note that the graph increases throughout the functionโ€™s domain, and as ๐‘ฅ approaches infinity, the outputs will also approach infinity, since the base, 10, is greater than one.

Now we can plot the line ๐‘ฆ=๐‘ฅ and reflect the graph of ๐‘ฆ=10๏— in it to produce the graph of the logarithmic function ๐‘ฆ=๐‘ฅlog๏Šง๏Šฆ (or, simply, ๐‘ฆ=๐‘ฅlog).

Reflecting the ๐‘ฆ-intercept of (0,1) through the line ๐‘ฆ=๐‘ฅ from the graph of the exponential function gives us an ๐‘ฅ-intercept of (1,0) on the graph of the logarithmic function. We can also reflect the point (1,10) to see that (10,1) lies on our logarithmic graph. Similarly, since the exponential function has a horizontal asymptote of the negative ๐‘ฅ-axis, the logarithmic function will have a vertical asymptote of the negative ๐‘ฆ-axis, with no portion of the graph lying to the left of the ๐‘ฆ-axis.

The domain and range of a function are the range and domain, respectively, of the inverse function. Hence, the logarithmic function has a domain of ]0,โˆž[ and a range of ]โˆ’โˆž,โˆž[. That is, the domain of the logarithmic function is the range of the exponential function, and the range of the logarithmic function is the domain of the exponential function. Finally, since the exponential function is an increasing function, its inverse will also be increasing.

Since the exponential curve will have similar properties if the base is any number greater than one, the logarithmic curve will also have a similar shape. Hence, we can use this to determine properties of the various logarithmic functions for ๐‘ฆ=๐‘ฅlog๏Š, where ๐‘›>1.

When we sketched our exponential curve, we used a base larger than 1, to create an increasing function. The base could also be between 0 and 1, so that the exponential function is decreasing.

Letโ€™s sketch the curve ๐‘ฆ=๐‘ฅlog๏Šฆ๏Ž–๏Šซ by reflecting the graph of ๐‘ฆ=0.5๏—. We will start by sketching ๐‘ฆ=0.5๏—.

The exponential function, 0.5๏—, has a domain of ]โˆ’โˆž,โˆž[ and a range of ]0,โˆž[. Its graph has the positive ๐‘ฅ-axis as an asymptote, with no portion of it lying below the ๐‘ฅ-axis, since 0.5 raised to a power can never be negative or 0. The graph has a ๐‘ฆ-intercept of 1, since 0.5=1๏Šฆ, and it passes through the point (1,0.5), since 0.5=0.5๏Šง. Also, note that the graph decreases throughout the functionโ€™s domain, and as ๐‘ฅ approaches negative infinity, the outputs will approach infinity, since the base is between zero and one.

We can then reflect this graph through the line ๐‘ฆ=๐‘ฅ to sketch the curve ๐‘ฆ=๐‘ฅlog๏Šฆ๏Ž–๏Šซ.

Reflecting the ๐‘ฆ-intercept of (0,1) and the point (1,0.5) through the line ๐‘ฆ=๐‘ฅ gives us an ๐‘ฅ-intercept of (1,0) and the point (0.5,1). Similarly, since the graph of the exponential function lies above the positive ๐‘ฅ-axis as a horizontal asymptote, the graph of the logarithmic function will lie to the right of the positive ๐‘ฆ-axis as a vertical asymptote.

Again, the logarithmic function has a domain of ]0,โˆž[ and a range of ]โˆ’โˆž,โˆž[. That is, the domain of the logarithmic function is the range of its inverse exponential function, and the range of the logarithmic function is the domain of the exponential function. Finally, since 0.5๏— is a decreasing function, its inverse, log๏Šฆ๏Ž–๏Šซ๐‘ฅ, is also a decreasing function.

Since the exponential curve will have similar properties if the base is any number between zero and one, the logarithmic curve will also have a similar shape. Hence, we can use this to determine properties of the various logarithmic functions for ๐‘ฆ=๐‘ฅlog๏Š, where ๐‘›>0 and ๐‘›โ‰ 1.

Properties: Graph of the Logarithmic Function ๐‘ฆ = log๐‘›(๐‘ฅ)

All graphs of logarithmic functions of the form ๐‘ฆ=(๐‘ฅ)log๏Š, where ๐‘›>0 and ๐‘›โ‰ 1,

  • have only one ๐‘ฅ-intercept, at 1;
  • pass through the point (๐‘›,1);
  • have a vertical asymptote at ๐‘ฅ=0;
  • have a domain of ]0,โˆž[ and a range of ]โˆ’โˆž,โˆž[.

When ๐‘›>1,

  • the function is increasing,
  • the function has a vertical asymptote to the right of the negative ๐‘ฆ-axis.

When 0<๐‘›<1,

  • the function is decreasing,
  • the function has a vertical asymptote to the right of the positive ๐‘ฆ-axis.

This gives us enough information to utilize the inverse relationship between exponential and logarithmic functions and sketch the curve ๐‘ฆ=๐‘ฅlog๏Š for any positive value of ๐‘› not equal to one. Letโ€™s see a few examples.

Example 1: Finding the Values of a Logarithmic Function

Find the missing values for โ„Ž(๐‘ฅ)=๐‘ฅlog๏Šจ.

๐‘ฅโˆ’212
โ„Ž(๐‘ฅ)

Answer

It would be possible to simply enter the values of ๐‘ฅ into a calculator to evaluate log๏Šจ๐‘ฅ in each case, but we will instead use the inverse relationship between logarithms and exponents to answer this question.

For the logarithmic function โ„Ž(๐‘ฅ)=๐‘ฅlog๏Šจ, the inverse exponential function is โ„Ž(๐‘ฅ)=2๏Šฑ๏Šง๏—. If the point (๐‘ฅ,๐‘ฆ) satisfies the logarithmic function, then the point (๐‘ฆ,๐‘ฅ) satisfies the exponential function. If we let ๐‘ฆ=โ„Ž(๐‘ฅ), we can say ๐‘ฆ=๐‘ฅlog๏Šจ; then, we can also say ๐‘ฅ=2๏˜. Now we are looking for values of ๐‘ฆ that satisfy this second equation for the given values of ๐‘ฅ.

First, substituting ๐‘ฅ=โˆ’2 gives us โˆ’2=2.๏˜

We know that there is no power to which we can raise the base of any exponential function that will give us either 0 or a negative number. This means that we cannot evaluate the logarithmic function at ๐‘ฅ=โˆ’2. Thus, โ„Ž(โˆ’2) is undefined.

Next, substituting ๐‘ฅ=1 into the rule ๐‘ฅ=2๏˜ gives us 1=2.๏˜

Remember, the base of any exponential function raised to the power 0 is equal to 1. In other words, since 2=1๏Šฆ, it follows that log๏Šจ1=0. Thus, โ„Ž(1)=0.

Finally, substituting ๐‘ฅ=2 into the rule ๐‘ฅ=2๏˜ gives us 2=2.๏˜

Recall that the base of any exponential function raised to the power 1 is equal to itself. In other words, since 2=2๏Šง, it follows that log๏Šจ2=1. Thus, โ„Ž(2)=1.

Therefore, after filling in the missing values in the table, we have the following:

๐‘ฅโˆ’212
โ„Ž(๐‘ฅ)Undefined01

Note

Although the question does not require this, we can sketch a graph of the logarithmic function โ„Ž(๐‘ฅ)๐‘ฅlog๏Šจ by first graphing the exponential function โ„Ž(๐‘ฅ)=2๏Šฑ๏Šง๏— and then reflecting this graph in the line ๐‘ฆ=๐‘ฅ. This is because the two functions are inverses. The graphs of both ๐‘ฆ=โ„Ž(๐‘ฅ) and ๐‘ฆ=โ„Ž(๐‘ฅ)๏Šฑ๏Šง are shown below.

We can see that the graph of the logarithmic function agrees with the answers we found.

Next, we will look at a problem in which we must identify the graph of a logarithmic function with a certain base.

Example 2: Graphing a Logarithmic Function with a Specified Base

Which graph represents the function ๐‘“(๐‘ฅ)=(๐‘ฅ)log๏Šซ?

Answer

The function ๐‘“(๐‘ฅ)=(๐‘ฅ)log๏Šซ is a logarithmic function of the form ๐‘“(๐‘ฅ)=๐‘ฅlog๏Š. We recall the following features for the graphs of such functions for any real value of ๐‘› greater than 0, such that ๐‘›โ‰ 1:

  • The graph has the ๐‘ฆ-axis as an asymptote, with no portion of the graph lying to the left of the ๐‘ฆ-axis.
  • The graph has an ๐‘ฅ-intercept of 1. This is because log๏Š1=0 or, inversely, ๐‘›=1๏Šฆ.
  • The graph passes through the point (๐‘›,1), since log๏Š๐‘›=1. In our case, ๐‘›=5, so this means the graph passes through the point (5,1).
  • If ๐‘›>1, the graph increases throughout the functionโ€™s domain. In our case, ๐‘›=5, so this is true.

We can see that all of the given graphs have the ๐‘ฆ-axis as an asymptote, with no portion of the graph lying to the left of the ๐‘ฆ-axis, and they are all increasing.

However, only one of the graphs has an ๐‘ฅ-intercept of 1 and passes through the point (5,1). The graph that represents the function ๐‘“(๐‘ฅ)=(๐‘ฅ)log๏Šซ is shown below.

We will now identify the graph of another logarithmic function. This time, all of the answer choices will be shown on the same coordinate plane.

Example 3: Recognizing the Graph of a Specified Logarithmic Function

Which curve is ๐‘ฆ=๐‘ฅlog๏Šฉ?

Answer

The curve ๐‘ฆ=(๐‘ฅ)log๏Šฉ is the graph of a logarithmic function of the form ๐‘“(๐‘ฅ)=๐‘ฅlog๏Š. We recall the following features for the graphs of such functions for any real value of ๐‘› greater than 0, such that ๐‘›โ‰ 1:

  • The graph has the ๐‘ฆ-axis as an asymptote, with no portion of the graph lying to the left of the ๐‘ฆ-axis.
  • The graph has an ๐‘ฅ-intercept of 1. This is because log๏Š1=0 or, inversely, ๐‘›=1๏Šฆ.
  • The graph passes through the point (๐‘›,1), since log๏Š๐‘›=1 or, inversely, ๐‘›=๐‘›๏Šง. In our case, since ๐‘›=3, this means the graph passes through the point (3,1).
  • If ๐‘›>1, the graph increases throughout the functionโ€™s domain. In our case, ๐‘›=3, so this is true.

We can see that all of the given graphs have the ๐‘ฆ-axis as an asymptote, with no portion of the graph lying to the left of the ๐‘ฆ-axis, and all of the graphs have an ๐‘ฅ-intercept of 1.

However, only two of the graphs are increasing across their domain, and only one passes through the point (3,1). The graph that represents the curve ๐‘ฆ=(๐‘ฅ)log๏Šฉ is (a).

We will not always be working with logarithmic functions on their own; sometimes we will be dealing with composite functions. By recalling our knowledge of function transformations, we can transform the graph of a logarithmic function by translation, dilation, and reflection. Each of these transformations has a different effect on the graph. Letโ€™s look at the graphs of some transformations of ๐‘“(๐‘ฅ)=๐‘ฅlog.

We will begin by looking at the graphs of ๐‘“(๐‘ฅ+โ„Ž)=(๐‘ฅ+โ„Ž)log when โ„Ž=2 and when โ„Ž=โˆ’2. Recall that these are horizontal translations of ๐‘“(๐‘ฅ)=๐‘ฅlog.

When โ„Ž=2, we have a translation of the graph of ๐‘ฆ=(๐‘ฅ)log by 2 units to the left. This also translates the vertical asymptote and the ๐‘ฅ-intercept 2 units to the left. Recall that the logarithm base is 10, so the graph of ๐‘ฆ=(๐‘ฅ)log๏Šง๏Šฆ passes through the point (10,1), and this point, too, is translated 2 units to the left. Note that this translation has also increased the domain from ]0,โˆž[ to ]โˆ’2,โˆž[, while the range of ]โˆ’โˆž,โˆž[ remains unchanged.

When โ„Ž=โˆ’2, we have a translation of the graph of ๐‘ฆ=(๐‘ฅ)log by 2 units to the right. This also translates the vertical asymptote, the ๐‘ฅ-intercept, and the point (10,1) by 2 units to the right. It reduces the domain from ]0,โˆž[ to ]2,โˆž[, while the range of ]โˆ’โˆž,โˆž[ remains unchanged.

These properties generalize to horizontal translations of logarithms of the form log๏Š๐‘ฅ, where ๐‘›>1. There are analogous cases for horizontal translations where the value of the logarithm base, ๐‘›, lies in 0<๐‘›<1. We will briefly consider the graphs of ๐‘“(๐‘ฅ+โ„Ž)=(๐‘ฅ+โ„Ž)log๏Šฆ๏Ž–๏Šซ when โ„Ž=2 and when โ„Ž=โˆ’2.

When โ„Ž=2, again we have a translation of the graph of ๐‘ฆ=(๐‘ฅ)log๏Šฆ๏Ž–๏Šซ by 2 units to the left. This also translates the vertical asymptote and the ๐‘ฅ-intercept by 2 units to the left. The graph of ๐‘ฆ=(๐‘ฅ)log๏Šฆ๏Ž–๏Šซ passes through the point (0.5,1), and this point is also translated by 2 units to the left. Note that this translation has also increased the domain from ]0,โˆž[ to ]โˆ’2,โˆž[, while the range of ]โˆ’โˆž,โˆž[ remains unchanged.

When โ„Ž=โˆ’2, we again have a translation of the graph of ๐‘ฆ=(๐‘ฅ)log๏Šฆ๏Ž–๏Šซ, the vertical asymptote, the ๐‘ฅ-intercept, and the point (0.5,1) by 2 units to the right. It reduces the domain from ]0,โˆž[ to ]2,โˆž[, while the range of ]โˆ’โˆž,โˆž[ remains unchanged.

These properties generalize to horizontal translations of logarithms of the form log๏Š๐‘ฅ, where 0<๐‘›<1.

Properties: Graphs of Horizontal Translations of the Logarithmic Function

In general, with a horizontal translation of the logarithmic function of the form ๐‘“(๐‘ฅ+โ„Ž)=(๐‘ฅ+โ„Ž)log๏Š, where ๐‘›>0 but ๐‘›โ‰ 1,

  • if โ„Ž>0, then the graph of ๐‘ฆ=๐‘“(๐‘ฅ+โ„Ž) is translated โ„Ž units to the left of ๐‘ฆ=(๐‘ฅ)log๏Š;
  • if โ„Ž<0, then the graph of ๐‘ฆ=๐‘“(๐‘ฅ+โ„Ž) is translated |โ„Ž| units to the right of ๐‘ฆ=(๐‘ฅ)log๏Š;
  • the domain of log๏Š(๐‘ฅ) is ]0,โˆž[, but the domain of ๐‘“(๐‘ฅ+โ„Ž) is ]โˆ’โ„Ž,โˆž[;
  • the range of log๏Š(๐‘ฅ) is ]โˆ’โˆž,โˆž[, and so is the range of ๐‘“(๐‘ฅ+โ„Ž);
  • if ๐‘›>1, then the function ๐‘“(๐‘ฅ+โ„Ž) increases throughout its domain, and its graph rises;
  • if 0<๐‘›<1, then the function ๐‘“(๐‘ฅ+โ„Ž) decreases throughout its domain, and its graph decreases;
  • the graph of ๐‘ฆ=๐‘“(๐‘ฅ+โ„Ž) has a vertical asymptote at ๐‘ฅ=โˆ’โ„Ž;
  • the graph of ๐‘ฆ=๐‘“(๐‘ฅ+โ„Ž) has an ๐‘ฅ-intercept of โˆ’โ„Ž+1;
  • the graph of ๐‘ฆ=๐‘“(๐‘ฅ+โ„Ž) passes through the point (๐‘›โˆ’โ„Ž,1).

Now let us consider some vertical translations of ๐‘“(๐‘ฅ)=๐‘ฅlog. We will look at the graphs of ๐‘“(๐‘ฅ)+๐‘˜=(๐‘ฅ)+๐‘˜log when ๐‘˜=2 and when ๐‘˜=โˆ’2.

When ๐‘˜=2, the graph of ๐‘ฆ=(๐‘ฅ)log is translated by 2 units upward. The vertical asymptote remains unchanged. The domain of ]0,โˆž[ remains unchanged, as does the range of ]โˆ’โˆž,โˆž[.

When ๐‘˜=โˆ’2, the graph of ๐‘ฆ=(๐‘ฅ)log is translated by 2 units downward. The vertical asymptote remains unchanged. The domain of ]0,โˆž[ remains unchanged, as does the range of ]โˆ’โˆž,โˆž[.

These properties generalize to vertical translations of logarithms of the form log๏Š๐‘ฅ, where ๐‘›>1. Again, there are analogous cases for vertical translations where the value of the logarithm base, ๐‘›, lies in 0<๐‘›<1.

Properties: Graphs of Vertical Translations of the Logarithmic Function

In general, with a vertical translation of the logarithmic function of the form ๐‘“(๐‘ฅ)+๐‘˜=(๐‘ฅ)+๐‘˜log๏Š, where ๐‘›>0 but ๐‘›โ‰ 1,

  • if ๐‘˜>0, then the graph of ๐‘ฆ=๐‘“(๐‘ฅ)+๐‘˜ is translated ๐‘˜ units upward from the graph of ๐‘ฆ=(๐‘ฅ)log๏Š;
  • if ๐‘˜<0, then the graph of ๐‘ฆ=๐‘“(๐‘ฅ)+๐‘˜ is translated |๐‘˜| units downward from the graph of ๐‘ฆ=(๐‘ฅ)log๏Š;
  • the domain of log๏Š(๐‘ฅ) is ]0,โˆž[, and so is the domain of ๐‘“(๐‘ฅ)+๐‘˜;
  • the range of log๏Š(๐‘ฅ) is ]โˆ’โˆž,โˆž[, and so is the range of ๐‘“(๐‘ฅ)+๐‘˜;
  • if ๐‘›>1, then the function ๐‘“(๐‘ฅ)+๐‘˜ increases throughout its domain, and its graph rises;
  • if 0<๐‘›<1, then the function ๐‘“(๐‘ฅ)+๐‘˜ decreases throughout its domain, and its graph decreases;
  • the graph of ๐‘ฆ=๐‘“(๐‘ฅ)+๐‘˜ has the ๐‘ฆ-axis as a vertical asymptote, the same as ๐‘ฆ=(๐‘ฅ)log๏Š.
  • the graph of ๐‘ฆ=๐‘“(๐‘ฅ)+๐‘˜ has a different ๐‘ฅ-intercept than the graph of ๐‘ฆ=(๐‘ฅ)log๏Š;
  • the graph of ๐‘ฆ=๐‘“(๐‘ฅ)+๐‘˜ passes through the point (๐‘›,1+๐‘˜).

Next, letโ€™s look at the graphs of ๐‘Žโ‹…๐‘“(๐‘ฅ)=๐‘Žโ‹…(๐‘ฅ)log when ๐‘Ž=0.5 and when ๐‘Ž=2. Recall that these are vertical stretches of ๐‘“(๐‘ฅ)=๐‘ฅlog, although some people refer to the first case as a compression, because the fractional scale factor results in the curve being compressed toward the ๐‘ฅ-axis.

This time our graphs show that when ๐‘Ž is 2, we have a vertical stretch of the graph of ๐‘ฆ=(๐‘ฅ)log by a scale factor of 2, and when ๐‘Ž is 0.5, we have a vertical stretch by a scale factor of 0.5.

These properties generalize to vertical dilations of logarithms of the form log๏Š๐‘ฅ, where ๐‘›>1, and there are analogous cases for vertical dilations where the value of the logarithm base, ๐‘›, lies in 0<๐‘›<1.

Properties: Graphs of Positive Vertical Dilations of the Logarithmic Function

In general, with a vertical stretch of the logarithmic function of the form ๐‘Žโ‹…๐‘“(๐‘ฅ)=๐‘Žโ‹…(๐‘ฅ)log๏Š, where ๐‘Ž is positive and ๐‘›>0 but ๐‘›โ‰ 1,

  • the graph of ๐‘ฆ=(๐‘ฅ)log๏Š is vertically stretched by a scale factor of ๐‘Ž to produce the graph of ๐‘ฆ=๐‘Žโ‹…๐‘“(๐‘ฅ);
  • the domain of log๏Š(๐‘ฅ) is ]0,โˆž[, and so is the domain of ๐‘Žโ‹…๐‘“(๐‘ฅ);
  • the range of log๏Š(๐‘ฅ) is ]โˆ’โˆž,โˆž[, and so is the range of ๐‘Žโ‹…๐‘“(๐‘ฅ);
  • if ๐‘›>1, then, since ๐‘Ž is positive, the function ๐‘Žโ‹…๐‘“(๐‘ฅ) increases throughout its domain, and its graph rises;
  • if 0<๐‘›<1, then, since ๐‘Ž is positive, the function ๐‘Žโ‹…๐‘“(๐‘ฅ) decreases throughout its domain, and its graph decreases;
  • the graph of ๐‘ฆ=(๐‘ฅ)log๏Š has the ๐‘ฆ-axis as a vertical asymptote, as does the transformed graph ๐‘ฆ=๐‘Žโ‹…๐‘“(๐‘ฅ);
  • the transformed graph ๐‘ฆ=๐‘Žโ‹…๐‘“(๐‘ฅ) has the same ๐‘ฅ-intercept, (1,0), as ๐‘ฆ=(๐‘ฅ)log๏Š;
  • the graph of ๐‘ฆ=๐‘Žโ‹…๐‘“(๐‘ฅ) passes through the point (๐‘›,๐‘Ž).

It is also important to note that if ๐‘Ž is negative, the graph is reflected through the ๐‘ฅ-axis. We can see this in the graphs of ๐‘ฆ=โˆ’2(๐‘ฅ)log and ๐‘ฆ=โˆ’0.5(๐‘ฅ)log below.

Properties: Graphs of Negative Vertical Dilations of the Logarithmic Function

In general, with a vertical stretch of the logarithmic function of the form ๐‘Žโ‹…๐‘“(๐‘ฅ)=๐‘Žโ‹…(๐‘ฅ)log๏Š, where ๐‘Ž is negative and ๐‘›>0 but ๐‘›โ‰ 1,

  • the graph of ๐‘ฆ=(๐‘ฅ)log๏Š is vertically stretched by a scale factor of ๐‘Ž and reflected in the ๐‘ฅ-axis to produce the graph of ๐‘ฆ=๐‘Žโ‹…๐‘“(๐‘ฅ);
  • the domain of log๏Š(๐‘ฅ) is ]0,โˆž[, and so is the domain of ๐‘Žโ‹…๐‘“(๐‘ฅ);
  • the range of log๏Š(๐‘ฅ) is ]โˆ’โˆž,โˆž[, and so is the range of ๐‘Žโ‹…๐‘“(๐‘ฅ);
  • if ๐‘›>1, then, since ๐‘Ž is negative, the function ๐‘Žโ‹…๐‘“(๐‘ฅ) decreases throughout its domain, and its graph decreases;
  • if 0<๐‘›<1, then, since ๐‘Ž is negative, the function ๐‘Žโ‹…๐‘“(๐‘ฅ) increases throughout its domain, and its graph rises;
  • the graph of ๐‘ฆ=(๐‘ฅ)log๏Š has the ๐‘ฆ-axis as a vertical asymptote, as does the transformed graph ๐‘ฆ=๐‘Žโ‹…๐‘“(๐‘ฅ);
  • the transformed graph ๐‘ฆ=๐‘Žโ‹…๐‘“(๐‘ฅ) has the same ๐‘ฅ-intercept, (1,0), as ๐‘ฆ=(๐‘ฅ)log๏Š;
  • the graph of ๐‘ฆ=๐‘Žโ‹…๐‘“(๐‘ฅ) passes through the point (๐‘›,๐‘Ž).

Finally, letโ€™s look at the graphs of ๐‘“(๐‘โ‹…๐‘ฅ)=(๐‘โ‹…๐‘ฅ)log when ๐‘=0.5 and when ๐‘=2. Recall that these are horizontal stretches of ๐‘“(๐‘ฅ)=๐‘ฅlog, although some people refer to the case where a fractional scale factor results in the curve being compressed toward the ๐‘ฆ-axis as a compression.

Here our graphs show that when ๐‘=2, we have a horizontal stretch of the graph of the logarithmic function by a scale factor of 0.5, and when ๐‘=0.5, we have a horizontal stretch by a scale factor of 2.

These properties generalize to horizontal dilations of logarithms of the form log๏Š๐‘ฅ, where ๐‘›>1, and there are analogous cases for horizontal dilations where the value of the logarithm base, ๐‘›, lies in 0<๐‘›<1.

Properties: Graphs of Positive Horizontal Dilations of the Logarithmic Function

In general, with a horizontal stretch of the logarithmic function of the form ๐‘“(๐‘โ‹…๐‘ฅ)=(๐‘โ‹…๐‘ฅ)log๏Š, where ๐‘ is positive and ๐‘›>0 but ๐‘›โ‰ 1,

  • the graph of ๐‘ฆ=(๐‘ฅ)log๏Š is horizontally stretched by a scale factor of 1๐‘ to produce the graph of ๐‘ฆ=๐‘“(๐‘โ‹…๐‘ฅ);
  • while the graph of ๐‘ฆ=(๐‘ฅ)log๏Š passes through the point (๐‘›,1), the graph of ๐‘ฆ=(๐‘โ‹…๐‘ฅ)log๏Š passes through the point ๏€ป๐‘›๐‘,1๏‡;
  • the domain of log๏Š(๐‘ฅ) is ]0,โˆž[, and so is the transformed domain of ๐‘“(๐‘โ‹…๐‘ฅ);
  • the range of log๏Š(๐‘ฅ) is ]โˆ’โˆž,โˆž[, and so is the transformed range of ๐‘“(๐‘โ‹…๐‘ฅ);
  • if ๐‘›>1, then, since ๐‘ is positive, the function ๐‘“(๐‘โ‹…๐‘ฅ) increases throughout its domain, and its graph rises;
  • if 0<๐‘›<1, then, since ๐‘ is positive, the function ๐‘“(๐‘โ‹…๐‘ฅ) decreases throughout its domain, and its graph decreases;
  • the graph of ๐‘ฆ=(๐‘ฅ)log๏Š has the ๐‘ฆ-axis as a vertical asymptote, as does the horizontally dilated graph ๐‘ฆ=๐‘“(๐‘โ‹…๐‘ฅ);
  • the graph of ๐‘ฆ=๐‘“(๐‘โ‹…๐‘ฅ) has an ๐‘ฅ-intercept of 1๐‘, which is different (if ๐‘โ‰ 1) than the ๐‘ฅ-intercept of (1,0) that ๐‘ฆ=(๐‘ฅ)log๏Š has.

It is also important to note that if ๐‘ is negative, the graph is reflected through the ๐‘ฆ-axis. We can see this in the graphs of ๐‘ฆ=(โˆ’2๐‘ฅ)log and ๐‘ฆ=(โˆ’0.5๐‘ฅ)log below.

Here our graphs show that when ๐‘=โˆ’2, we have a horizontal stretch of the graph of the logarithmic function by a scale factor of 0.5, but the graph is also reflected in the ๐‘ฆ-axis. When ๐‘=โˆ’0.5, we have a horizontal stretch by a scale factor of 2, and again, the graph is reflected in the ๐‘ฆ-axis.

These properties generalize to horizontal dilations of logarithms of the form log๏Š๐‘ฅ, where ๐‘›>1, and there are analogous cases for horizontal dilations where the value of the logarithm base, ๐‘›, lies in 0<๐‘›<1.

Properties: Graphs of Negative Horizontal Dilations of the Logarithmic Function

In general, with a horizontal stretch of the logarithmic function of the form ๐‘“(๐‘โ‹…๐‘ฅ)=(๐‘โ‹…๐‘ฅ)log๏Š, where ๐‘ is negative and ๐‘›>0 but ๐‘›โ‰ 1,

  • the graph of ๐‘ฆ=(๐‘ฅ)log๏Š is horizontally stretched by a scale factor of 1๐‘ to produce the graph of ๐‘ฆ=๐‘“(๐‘โ‹…๐‘ฅ);
  • while the graph of ๐‘ฆ=(๐‘ฅ)log๏Š passes through the point (๐‘›,1), the graph of ๐‘ฆ=(๐‘โ‹…๐‘ฅ)log๏Š passes through the point ๏€ป๐‘›๐‘,1๏‡;
  • the graph of ๐‘ฆ=๐‘“(๐‘โ‹…๐‘ฅ) is a reflection in the ๐‘ฆ-axis of ๐‘ฆ=๐‘“(|๐‘|โ‹…๐‘ฅ);
  • the domain of log๏Š(๐‘ฅ) is ]0,โˆž[, but the transformed domain of ๐‘“(๐‘โ‹…๐‘ฅ) is ]โˆ’โˆž,0[;
  • the range of log๏Š(๐‘ฅ) is ]โˆ’โˆž,โˆž[, and so is the transformed range of ๐‘“(๐‘โ‹…๐‘ฅ);
  • if ๐‘›>1, then, since ๐‘ is negative, the function ๐‘“(๐‘โ‹…๐‘ฅ) decreases throughout its domain, and its graph decreases;
  • if 0<๐‘›<1, then, since ๐‘ is negative, the function ๐‘“(๐‘โ‹…๐‘ฅ) increases throughout its domain, and its graph rises;
  • the graph of ๐‘ฆ=(๐‘ฅ)log๏Š has the ๐‘ฆ-axis as a vertical asymptote, as does the horizontally dilated graph ๐‘ฆ=๐‘“(๐‘โ‹…๐‘ฅ);
  • the graph of ๐‘ฆ=๐‘“(๐‘โ‹…๐‘ฅ) has an ๐‘ฅ-intercept of 1๐‘, which is different than the ๐‘ฅ-intercept of (1,0) that ๐‘ฆ=(๐‘ฅ)log๏Š has.

It is also worth noting at this point that with transformations of the form ๐‘“(๐‘โ‹…๐‘ฅ)=(๐‘โ‹…๐‘ฅ)log on the logarithmic function ๐‘“(๐‘ฅ)=(๐‘ฅ)log, we can rearrange the expression for ๐‘“(๐‘โ‹…๐‘ฅ) using the product rule of logarithms if ๐‘ is positive: logloglog๐‘“(๐‘โ‹…๐‘ฅ)=(๐‘)+(๐‘ฅ).

Since ๐‘ is a constant, we can let ๐‘˜=(๐‘)log and rewrite the transformed function as loglog๐‘“(๐‘โ‹…๐‘ฅ)=(๐‘ฅ)+๐‘˜.

Hence we can say ๐‘“(๐‘โ‹…๐‘ฅ)=๐‘“(๐‘ฅ)+๐‘˜.

We recall that this describes a vertical translation of the function ๐‘“(๐‘ฅ) by ๐‘˜ units. Just like horizontal stretching or compression of the logarithmic function, a vertical translation leaves the functionโ€™s domain, range, and vertical asymptote unchanged, and the graph rises throughout the functionโ€™s domain, but the ๐‘ฅ-intercept changes. It is a feature of the logarithmic function that these transformations are equivalent.

Now, letโ€™s use this information about transformed logarithmic functions and their graphs to solve some more problems.

In our next problem, we are given the graph of a logarithmic function and are asked to determine which function the graph represents. We will see that we can use the concept of transformations to help us find the correct answer.

Example 4: Recognizing Differences between Transformations of Logarithmic Graphs

Which function represents the following graph?

  1. ๐‘“(๐‘ฅ)=(2๐‘ฅ)log๏Šช
  2. ๐‘“(๐‘ฅ)=(๐‘ฅ)log๏Šช
  3. ๐‘“(๐‘ฅ)=(2๐‘ฅ)log๏Šจ
  4. ๐‘“(๐‘ฅ)=(๐‘ฅ)log๏Šจ
  5. ๐‘“(๐‘ฅ)=(๐‘ฅ)log๏Šฎ

Answer

Our initial observation of the graph tells us that the shape resembles that of the graph of a logarithmic function. There is a vertical asymptote at ๐‘ฅ=0; no part of the graph is to the left of the asymptote; and the function is increasing. The domain appears to be ]0,โˆž[, and the range appears to be ]โˆ’โˆž,โˆž[. Indeed, examining the options given for the answer, we are being asked to find which transformation of a logarithmic function fits the graph.

If this function were of the form ๐‘“(๐‘ฅ)=(๐‘ฅ)log๏Š, with ๐‘›>0 and ๐‘›โ‰ 1, then the graph would have an ๐‘ฅ-intercept of 1 and pass through the point (๐‘›,1). However, the ๐‘ฅ-intercept is at 12, so it looks like this is a transformation of a simple logarithmic function.

Because the vertical asymptote remains at ๐‘ฅ=0, this could be one of the following transformations:

  • A horizontal stretch (or compression) of the form ๐‘“(๐‘ฅ)=(๐‘โ‹…๐‘ฅ)log๏Š, in which case the ๐‘ฅ-intercept would be at 1๐‘, and the graph would pass through the point ๏€ป๐‘›๐‘,1๏‡
  • A vertical translation of the form ๐‘“(๐‘ฅ)=(๐‘ฅ)+๐‘˜log๏Š

This could not be a horizontal translation, as that would change the position of the vertical asymptote. It also could not be a vertical stretch (or compression) of the form ๐‘“(๐‘ฅ)=๐‘Žโ‹…(๐‘ฅ)log๏Š, as this would still have an ๐‘ฅ-intercept of 1. However, it is possible that the transformation could be a combination of a vertical stretch with either a horizontal stretch, a vertical translation, or both. Looking at the options given in the question, we do not need to consider such combinations of transformations.

The graph we have been given has an ๐‘ฅ-intercept of 12 and passes through the point (2,1). If it is a horizontal stretch of the form ๐‘“(๐‘ฅ)=(๐‘โ‹…๐‘ฅ)log๏Š, then, from the intercept, we could say that 1๐‘=12.

This means that ๐‘=2. Furthermore, from the ๐‘ฅ-coordinate when ๐‘ฆ=1, we could say that ๐‘›๐‘=2, but we know that ๐‘=2, so we can substitute this value into the above equation and rearrange to find that ๐‘›=4.

This would make our transformed logarithmic function ๐‘“(๐‘ฅ)=(2๐‘ฅ)log๏Šช.

This function can also be presented as a vertical translation, using the product rule of logarithms: loglogloglog๏Šช๏Šช๏Šช๏Šช(2โ‹…๐‘ฅ)=(2)+(๐‘ฅ)=(๐‘ฅ)+12.

An alternative way to present our transformed logarithmic function is ๐‘“(๐‘ฅ)=(๐‘ฅ)+12log๏Šช.

Looking at the options given, we see that our answer is A, ๐‘“(๐‘ฅ)=(2๐‘ฅ)log๏Šช.

Finally, letโ€™s look at an example in which we must determine the values of multiple logarithmic expressions from the graph of an exponential function. We will need to keep in mind that every logarithmic function is the inverse of an exponential function.

Example 5: Finding Values of a Logarithm from a Graph

Use the graph of ๐‘ฆ=10๏— to list the values of log๏Š for ๐‘›=2,โ€ฆ,6 to two decimal places. For example, we see that log2โ‰ˆ0.30.

Answer

Remember, a logarithmic function is the inverse of an exponential function. If the point (๐‘ฅ,๐‘ฆ) satisfies the exponential function, then the point (๐‘ฆ,๐‘ฅ) satisfies the logarithmic function. This means that since we are given the graph of the exponential function ๐‘ฆ=10๏—, we should look on the ๐‘ฆ-axis for the value we will substitute for ๐‘› in the expression log๐‘› instead of the ๐‘ฅ-axis. (Recall that, in the expression log๐‘›, a base of 10 is assumed, and we could write it as log๏Šง๏Šฆ๐‘›.)

For example, we can see that locating the point on the graph with a ๐‘ฆ-coordinate of 2 and then finding the ๐‘ฅ-coordinate of this point give a value of approximately 0.3 for the expression log2. Letโ€™s repeat this process to find approximate values for log3, log4, log5, and log6.

First, we will approximate the value of log3 by locating the point on the graph with a ๐‘ฆ-coordinate of 3 and then finding the ๐‘ฅ-coordinate of this point.

Since the ๐‘ฅ-coordinate is approximately 0.48, we can say that this is the value of log3 to two decimal places.

Next, letโ€™s approximate the value of log4 by locating the point on the graph with a ๐‘ฆ-coordinate of 4 and then finding the ๐‘ฅ-coordinate of this point.

The ๐‘ฅ-coordinate is about 0.60, so we know that this is the value of log4 to two decimal places.

Now we will approximate the value of log5 by locating the point on the graph with a ๐‘ฆ-coordinate of 5 and then finding the ๐‘ฅ-coordinate of this point.

Since the ๐‘ฅ-coordinate is about 0.70, we know that this is the value of log5 to two decimal places.

Finally, letโ€™s approximate the value of log6 by locating the point on the graph with a ๐‘ฆ-coordinate of 6 and then finding the ๐‘ฅ-coordinate of this point.

The ๐‘ฅ-coordinate is about 0.78, so we know that this is the value of log6 to two decimal places.

In summary, the values of log๐‘› for ๐‘›=2,โ€ฆ,6 to two decimal places are 0.30, 0.48, 0.60, 0.70, and 0.78.

Now letโ€™s finish by recapping some key points from this explainer.

Key Points

  • A logarithmic function is the inverse of an exponential function. For the exponential function ๐‘“(๐‘ฅ)=๐‘›๏—, where ๐‘›>0 and ๐‘›โ‰ 1, the inverse logarithmic function is ๐‘“(๐‘ฅ)=๐‘ฅ๏Šฑ๏Šง๏Šlog. If the point (๐‘ฅ,๐‘ฆ) satisfies the exponential function, then the point (๐‘ฆ,๐‘ฅ) satisfies the logarithmic function.
  • The graphs of an exponential function and its inverse logarithmic function are reflections in the line ๐‘ฆ=๐‘ฅ.
  • The graph of a logarithmic function can be translated, stretched, or reflected.
  • For any real number value of ๐‘› greater than 0, such that ๐‘›โ‰ 1, the graph of ๐‘”(๐‘ฅ)=๐‘ฅlog๏Š has the ๐‘ฆ-axis as an asymptote and an ๐‘ฅ-intercept of 1, with no portion of the graph lying to the left of the ๐‘ฆ-axis.
  • For any real number value of ๐‘› greater than 0, such that ๐‘›โ‰ 1, the graph of ๐‘”(๐‘ฅ)=๐‘ฅlog๏Š passes through the point (๐‘›,1). That is, log๏Š๐‘›=1.
  • If 0<๐‘›<1 for the function ๐‘”(๐‘ฅ)=๐‘ฅlog๏Š, the functionโ€™s graph decreases throughout its domain, and if ๐‘›>1, the graph increases throughout its domain.

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