Lesson Explainer: Converting between Logarithmic and Exponential Forms Mathematics

In this explainer, we will learn how to convert between logarithmic and exponential forms of equations.

The motivation for this is mathematical, but it also has real-world applications such as measuring sound using the decibel scale, which is used to compare signals in acoustics and electronics, the intensities of different earthquakes using the Richter scale, or the brightness of stars, to name a few.

These conversions give us equivalent forms which also allow us to solve exponential or logarithmic equations, with unknowns appearing as exponents or logarithms. For example, suppose we want to determine the value of ๐‘ฅ such that 10=600.๏—

So how do we solve for ๐‘ฅ? One method would be trial and error, but since 600 is not a power of 10, it cannot be an integer value. We know that 10=100๏Šจ and 10=1000๏Šฉ, and since ๐‘ฆ=10๏— is an increasing function, our ๐‘ฅ must be some value between 2 and 3. We can also graph the functions ๐‘ฆ=10๏— and ๐‘ฆ=600 and see where they intersect; this will give us an idea of what the ๐‘ฅ value should be.

A better way would be to get a more exact solution by converting this exponential equation into logarithmic form in order to make ๐‘ฅ the subject; by doing this we can immediately write ๐‘ฅ=600=2.778(4).logsf๏Šง๏Šฆ

We expect this value from our general consideration of the function ๐‘ฆ=10๏— for 2<๐‘ฅ<3.

Letโ€™s now remind ourselves how to express all exponential functions in logarithmic form and vice versa.

Definition: Relationship between Logarithmic and Exponential Forms

For ๐‘ฅ>0 and base ๐‘Ž>0, ๐‘Žโ‰ 1, the exponential form ๐‘ฆ=๐‘Ž๏— is equivalent to the logarithmic form ๐‘ฅ=๐‘ฆlog๏Œบ, which allows us to convert from one form to another once we identify ๐‘Ž, ๐‘ฅ, and ๐‘ฆ.

An exponential function ๐‘ฆ=๐‘Ž๏— with domain ]โˆ’โˆž,โˆž[ and range ]0,โˆž[ is the inverse of the logarithmic function ๐‘ฆ=๐‘ฅlog๏Œบ; thus, the domain and range of the logarithmic function are interchanged for the exponential function, given by ]0,โˆž[ and ]โˆ’โˆž,โˆž[ respectively.

Suppose we want to write 5=25๏Šจ in logarithmic form. The first step would be to compare this with ๐‘Ž=๐‘ฆ๏— and identify the constants ๐‘Ž, ๐‘ฅ, and ๐‘ฆ which in this case are given by ๐‘Ž=5, ๐‘ฅ=2, and ๐‘ฆ=25.

The equivalent logarithmic form is given by log๏Œบ๐‘ฆ=๐‘ฅ, which, once we substitute the values, can be written as log๏Šซ25=2.

Therefore, 5=25๏Šจ, in exponential form, is equivalent to log๏Šซ25=2 in logarithmic form.

We will now look at several examples to gain a deeper understanding of the relationship between these forms. First, letโ€™s look at a simple example on base ๐‘Ž=10 using an exact expression.

Example 1: Converting an Equation from Exponential to Logarithmic Form

Express 10=1000๏Šฉ in its equivalent logarithmic form.

Answer

In this example, we will use the equivalence between the exponential and logarithmic forms and use this to convert from exponential form to logarithmic form by identifying the variables which appear in the general form.

Recall that the exponential form ๐‘Ž=๐‘ฆ๏— is equivalent to the logarithmic form log๏Œบ๐‘ฆ=๐‘ฅ.

By comparing 10=1000๏Šฉ with the exponential form, we can identify our ๐‘Ž, ๐‘ฆ, and ๐‘ฅ as ๐‘Ž=10, ๐‘ฆ=1000, and ๐‘ฅ=3. Using this with the logarithmic form, we have the equivalent form loglog๏Šง๏Šฆ1000=1000=3, where we omit the base for log๏Šง๏Šฆ, which is the convention when the base, ๐‘Ž, is equal to 10. This is known as the common logarithm.

Now letโ€™s consider an example where we have to solve an equation to find an unknown value, ๐‘ฅ, using this method.

Example 2: Rewriting an Exponential Equation in Logarithmic Form

Write the exponential equation ๐‘’=5๏— in logarithmic form.

Answer

In this example, ๐‘ฅ is an unknown value which we can find by rewriting the expression in logarithmic form, making ๐‘ฅ the subject of the equation.

By comparing this form with the exponential function ๐‘Ž=๐‘ฆ๏—, we can identify our ๐‘Ž and ๐‘ฆ as ๐‘Ž=๐‘’ and ๐‘ฆ=5.

Using this with the logarithmic form ๐‘ฅ=๐‘ฆlog๏Œบ, we have the equivalent form ๐‘ฅ=5=5,logln๏Œพ where we write the base for log๏Œพ as ln, which is the convention when the base, ๐‘Ž, is equal to ๐‘’. This is known as the natural logarithm.

The next example is one with a different base from the common, ๐‘Ž=10, or natural, ๐‘Ž=๐‘’, negative powers and a fraction, although the procedure is exactly the same.

Example 3: Converting an Equation from Exponential to Logarithmic Form

Express 4=116๏Šฑ๏Šจ in its equivalent logarithmic form.

Answer

In this example, we have an exact expression that does not contain any variables for which we have to solve.

We want to convert this expression from exponential to logarithmic form.

By comparing this form with the exponential function ๐‘Ž=๐‘ฆ๏—, we can identify our ๐‘Ž, ๐‘ฆ, and ๐‘ฅ as ๐‘Ž=4, ๐‘ฆ=116, and ๐‘ฅ=โˆ’2. By comparing the exponential form with the logarithmic form, we have that 4=116๏Šฑ๏Šจ is equivalent to log๏Šช116=โˆ’2.

Now letโ€™s look at an example where we do the same thing in reverse, going from logarithmic to exponential form.

Example 4: Converting an Equation from Logarithmic to Exponential Form

Express log1000000=6 in its equivalent exponential form.

Answer

In this example, we have an exact expression which we need to convert from logarithmic to exponential form.

Recall that the logarithmic form, log๏Œบ๐‘ฆ=๐‘ฅ, is equivalent to ๐‘Ž=๐‘ฆ๏—. Remember that if the base of a logarithm is not given, we can assume that is has base 10.

Here, we have that ๐‘Ž=10, ๐‘ฆ=1000000, and ๐‘ฅ=6, and, therefore, log1000000=6 is equivalent to 10=1000000.๏Šฌ

In our final example, letโ€™s look at how to convert a logarithmic equation to exponential form.

Example 5: Rewriting a Logarithmic Equation in Exponential Form

Write the logarithmic equation 8=๐‘ฅln in exponential form.

Answer

In this example, ๐‘ฅ is an unknown value which we can find by rewriting the expression in exponential form and making ๐‘ฅ the subject of the equation.

Remember that if we see a natural logarithm ln๐‘ฅ, we can assume that is has base ๐‘’.

Here, we have that ๐‘Ž=๐‘’, and ๐‘ฆ=8, and, therefore, 8=๐‘ฅln is equivalent to ๐‘’=๐‘ฅ.๏Šฎ

Now, letโ€™s look at how we can use these conversions to solve a real-world problem. Suppose you want to compare the intensity, ๐ผ๏ƒ, or magnitudes, ๐‘€๏ƒ, of two different earthquakes with ๐‘–=1,2. Their relationship is given by the equation ๐ผ=๐ผ10,๏Šง๏Šจ๏Œฌ๏Šฑ๏Œฌ๏Ž ๏Žก which can be rewritten as ๐ผ๐ผ=10.๏Šง๏Šจ๏Œฌ๏Šฑ๏Œฌ๏Ž ๏Žก

The magnitudes, ๐‘€๏ƒ, are measured by a base-10 logarithmic scale known as the Richter scale.

Now suppose we have one earthquake with intensity ๐ผ๏Šง which has exactly 600 times the intensity of a second earthquake with intensity ๐ผ๏Šจ. Algebraically, this can be expressed as ๐ผ=600๐ผ;๏Šง๏Šจ hence ๐ผ๐ผ=600.๏Šง๏Šจ

We want to calculate the difference in magnitude of the earthquakes, which we denote by ๐‘ฅ=๐‘€โˆ’๐‘€๏Šง๏Šจ, and combining the two equations we have to solve 10=600.๏—

By converting this equation into logarithmic form, we can find the value of ๐‘ฅ as ๐‘ฅ=600=2.778(4).logsf๏Šง๏Šฆ

Another way to solve these types of problems with exponents, instead of having to compare the expressions with the logarithmic or exponential forms, is to take common or natural logs of both sides of the equation. A log of an arbitrary base ๐‘Ž can be expressed in terms of common or natural logs as loglnlnloglog๏Œบ๐‘ฆ=๐‘ฆ๐‘Ž=๐‘ฆ๐‘Ž.

Applying the common logs to both sides of the equation ๐‘ฆ=๐‘Ž๏—, we have logloglog๐‘ฆ=๐‘Ž=๐‘ฅ๐‘Ž,๏— which can be rearranged to give ๐‘ฅ=๐‘ฆ๐‘Ž=๐‘ฆ.logloglog๏Œบ

However, for this, you have to use the laws of logs or exponents, which is beyond the scope of this explainer and will be covered in more detail in another lesson.

Key Points

  • An exponential function ๐‘ฆ=๐‘Ž๏— is the inverse of the logarithmic function ๐‘ฆ=๐‘ฅlog๏Œบ.
  • The common logarithm has base 10 and is generally written as ๐‘ฆ=๐‘ฅlog and is equivalent to ๐‘ฅ=10๏˜.
  • The natural logarithm has base ๐‘’ and is generally written as ๐‘ฆ=๐‘ฅln and is equivalent to ๐‘ฅ=๐‘’๏˜.

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