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 such that
So how do we solve for ? 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 and , and since is an increasing function, our must be some value between 2 and 3. We can also graph the functions and and see where they intersect; this will give us an idea of what the 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 the subject; by doing this we can immediately write
We expect this value from our general consideration of the function for .
Let’s now remind ourselves how to express all exponential functions in logarithmic form and vice versa.
Definition: Relationship between Logarithmic and Exponential Forms
For and base , , the exponential form is equivalent to the logarithmic form , which allows us to convert from one form to another once we identify , , and .
An exponential function with domain and range is the inverse of the logarithmic function ; thus, the domain and range of the logarithmic function are interchanged for the exponential function, given by and respectively.
Suppose we want to write in logarithmic form. The first step would be to compare this with and identify the constants , , and which in this case are given by , , and .
The equivalent logarithmic form is given by , which, once we substitute the values, can be written as .
Therefore, , in exponential form, is equivalent to in logarithmic form.
We will now look at several examples to gain a deeper understanding of the relationship between these forms. First, let’s look at a simple example on base using an exact expression.
Example 1: Converting an Equation from Exponential to Logarithmic Form
Express in its equivalent logarithmic form.
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 is equivalent to the logarithmic form .
By comparing with the exponential form, we can identify our , , and as , , and . Using this with the logarithmic form, we have the equivalent form where we omit the base for , which is the convention when the base, , is equal to 10. This is known as the common logarithm.
Now let’s consider an example where we have to solve an equation to find an unknown value, , using this method.
Example 2: Rewriting an Exponential Equation in Logarithmic Form
Write the exponential equation in logarithmic form.
In this example, is an unknown value which we can find by rewriting the expression in logarithmic form, making the subject of the equation.
By comparing this form with the exponential function , we can identify our and as and .
Using this with the logarithmic form , we have the equivalent form where we write the base for as , which is the convention when the base, , is equal to . This is known as the natural logarithm.
The next example is one with a different base from the common, , or natural, , negative powers and a fraction, although the procedure is exactly the same.
Example 3: Converting an Equation from Exponential to Logarithmic Form
Express in its equivalent logarithmic form.
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 , we can identify our , , and as , , and . By comparing the exponential form with the logarithmic form, we have that is equivalent to
Now let’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 in its equivalent exponential form.
In this example, we have an exact expression which we need to convert from logarithmic to exponential form.
Recall that the logarithmic form, , is equivalent to . Remember that if the base of a logarithm is not given, we can assume that is has base 10.
Here, we have that , , and , and, therefore, is equivalent to
In our final example, let’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 in exponential form.
In this example, is an unknown value which we can find by rewriting the expression in exponential form and making the subject of the equation.
Remember that if we see a natural logarithm , we can assume that is has base .
Here, we have that , and , and, therefore, is equivalent to
Now, let’s look at how we can use these conversions to solve a real-world problem. Suppose you want to compare the intensity, , or magnitudes, , of two different earthquakes with . Their relationship is given by the equation which can be rewritten as
The magnitudes, , are measured by a base-10 logarithmic scale known as the Richter scale.
Now suppose we have one earthquake with intensity which has exactly 600 times the intensity of a second earthquake with intensity . Algebraically, this can be expressed as hence
We want to calculate the difference in magnitude of the earthquakes, which we denote by , and combining the two equations we have to solve
By converting this equation into logarithmic form, we can find the value of as
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 of an arbitrary base can be expressed in terms of common or natural logs as
Applying the common logs to both sides of the equation , we have which can be rearranged to give
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.
- An exponential function is the inverse of the logarithmic function .
- The common logarithm has base 10 and is generally written as and is equivalent to .
- The natural logarithm has base and is generally written as and is equivalent to .