In this explainer, we will learn how to use logarithms to solve exponential equations.
Let’s start by considering the exponential equation . We can see that 2 is being raised to the power of on the left side. In other words, a variable occurs in an exponent. This is a common feature of all exponential equations.
Definition: Exponential Equation
An exponential equation is an equation in which a variable is used in one exponent or more.
Before looking at how to solve using logarithms, let’s explore two alternative methods we might use to solve it. We should get the same solution using these methods as those we get when using logarithms.
The first alternative method is to begin by defining the functions and . We could then graph and on the same coordinate plane and determine the point of intersection of the graphs as shown.
The -coordinate of the point of intersection is 3, so with this method we get a solution of .
The second alternative method is to start by using the fact that 8 is a power of 2 to rewrite the equation. We know that , so we could substitute for 8 to get . Then we could equate the exponents to again get a solution of .
Now let’s consider how to solve by using the relationship between exponential and logarithmic functions.
Definition: Logarithmic Function
A logarithmic function is the inverse of an exponential function. If , then .
Since , or , is in the form , we know that the value of is 8 and that the value of is 2. Thus, we can write the equation . To simplify the right side, we can ask ourselves, “A base of 2 raised to what power is equal to 8?” The answer is 3, so we get , the same answer that we arrived at with the previous two methods.
Some exponential equations are more complex, however, so when solving them using logarithms, we often need to use one or more of the rules of logarithms below.
Properties: Rules of Logarithms
Product Rule:
Quotient Rule:
Power Rule:
Notice that, in each of the rules, the bases of the logarithms are the same on both sides of the equation.
- The product rule states that the of the product of two numbers is the sum of the of the first factor and the of the second factor. We would use the product rule to find that .
- The quotient rule states that the of the quotient of two numbers is the difference of the of the dividend and the of the divisor. We would use the quotient rule to find that .
- The power rule states that the of a base raised to a power is the product of the power and the of the base. We would use the power rule to find that .
When solving exponential equations using logarithms, we often use a base of 10 or a base of for the because of the buttons on our calculators. However, the base does not matter. Recall that when the base is 10, by convention, there is no need to specify it, and when the base is , we are taking the natural . It is important to note that if we use a base of 10 or a base of when taking the of a number, often the result will not be an integer. This is fine, though, because one of the advantages of solving an exponential equation using logarithms is that the equation does not have to have an integer solution. We can use a scientific calculator to approximate the solution. Let’s look at how we would do this in the examples that follow.
Example 1: Solving Exponential Equations Using Logarithms
Solve for , giving your answer to three decimal places.
Answer
Let’s begin by recalling that a logarithmic function is the inverse of an exponential function. If , then . Since , or , is in the form , we know that the value of is 11 and that the value of is 3. Thus, we can write the equation
Since 11 is not a power of 3, we need to use a scientific calculator to simplify the right-hand side. We must make sure to use the appropriate keystrokes, keeping in mind that the base is 3 and not 10. In doing so, we get , The problem asks for the value of to three decimal places, so we need to consider the digit in the ten thousandths place, which is 6. Because this digit is greater than or equal to 5, we should round the digit 2 in the thousands place up to get an answer of .
Another way to solve the equation is to take the of both sides. Using a base of 10 gives us
Notice that the base is not specified in the equation. The power rule of logarithms then allows us to rewrite as so that the equation becomes
Next, we can divide both sides by to get and then use the button on a scientific calculator to get .
Just as before, we would round the digit 2 in the thousandths place up to arrive at an answer of .
Note
When using a scientific calculator to approximate the value of the expression , it is important that we write both and to a sufficient number of decimal places if we calculate them separately. If, for example, we only wrote them to three decimal places, we would have got
In this case, we would have kept the 2 in the thousandths place instead of rounding it up, and we would have got an incorrect answer of . For this reason, it is best to enter into our calculator as a single expression rather than calculating each individual .
Next, we will work on solving a problem involving an exponential equation with a binomial exponent.
Example 2: Solving Exponential Equations with Binomial Exponents Using Logarithms
Find, to the nearest hundredth, the value of for which .
Answer
In order to solve the equation for , we can start by taking the of both sides. If we use a base of 10, we will not have to specify it and will get the equation
Using the power rule of logarithms, we can then rewrite as so that the equation becomes
Dividing both sides of the equation by gives us and after subtracting 8 from both sides we get
Now we can use the button on a scientific calculator to help approximate the value of . When doing so, it is best to enter into the calculator as a single expression rather than finding and separately before simplifying. This way, there is no danger of rounding errors. Doing this gives us which gives us an answer of to the nearest hundredth.
Now let’s look at how to solve an equation with two binomial exponents instead of one.
Example 3: Solving Exponential Equations with Binomial Exponents Using Logarithms
Use a calculator to find the value of for which . Give your answer correct to two decimal places.
Answer
To start, let’s take the of both sides of the equation. Using a base of 10, we get
The power rule of logarithms then allows us to rewrite as and as , which gives us the equation
After distributing on the left side of the equation and on the right side, the equation becomes
Now, in order to isolate the variable, let’s move the terms containing an to one side of the equation and the terms not containing an to the other side. First, we will add to both sides to get
Next, we will subtract from both sides so that the equation becomes
It is now possible for us to factor out an from the right side of the equation, giving us and after we divide both sides of the equation by the expression , we arrive at
Finally, we can use the button on a scientific calculator to enter the expression for . The calculator gives us which gives us an answer of to two decimal places.
In the example that follows, we must also move the variable terms to one side of the equation.
Example 4: Solving Exponential Equations Using Logarithms
Solve for , giving your answer to three decimal places.
Answer
The first step in solving for is moving the terms with an exponent of to one side of the equation and the terms without an exponent of to the other side. If we divide both sides of the equation by , we get
We can see that there is now a 2 in both the numerator and the denominator of the fraction on the left side of the equation and a in both the numerator and the denominator of the fraction on the right side. These terms will cancel out, giving us
Recall that if two constants and are both raised to the power of , then ; so we can replace with , getting
Because a logarithmic function is the inverse of an exponential function, we know that if , then . The equation , or , is in the form , which tells us that the value of is and that the value of is . Thus, we can write the equation
Now we can use a scientific calculator to simplify the right side, keeping in mind that the base is and not 10. In doing so, we get , which gives us an answer of to three decimal places.
We can also take the of both sides of the equation to solve it. If we use a base of 10, we get
Using the power rule of logarithms, we can rewrite as , giving us the equation
Now, we can divide both sides by to get and then use the button on a scientific calculator to enter the expression for , giving us
Again, we would keep the digit 5 in the thousandths place to arrive at an answer of .
Finally, let’s look at an example in which we must use two different rules of logarithms.
Example 5: Solving Exponential Equations with Binomial Exponents Using Logarithms
Use a calculator to find the value of for which . Give your answer correct to two decimal places.
Answer
Let’s begin by taking the log of both sides of the equation. Using a base of 10 gives us
The product rule of logarithms allows us to rewrite as and as , which results in the equation
We can now use the power rule of logarithms to rewrite as . We can also use it to rewrite as , giving us the equation
After we then distribute on the right side, we get
Next, we must move the terms containing an to one side of the equation and the terms not containing an to the other side. Subtracting from both sides gives us and then subtracting from both sides gives us
The last two terms on the right side can now be combined to get the equation
It is now possible for us to factor out an from the left side of the equation, giving us and after we divide both sides of the equation by the expression , we arrive at
Finally, we can use the button on a scientific calculator to enter the expression for . The calculator gives us which gives us an answer of to two decimal places.
Now let’s finish by recapping some key points.
Key Points
- An exponential equation is an equation in which a variable is used in one exponent or more.
- A logarithmic function is the inverse of an exponential function. If , then .
- When solving exponential equations using logarithms, we must often use one or more of the rules of logarithms. Three logarithmic rules that are used when solving exponential equations are the product rule, the quotient rule, and the power rule.
- The product rule, or , states that the of the product of two numbers is the sum of the of the first factor and the of the second factor.
- The quotient rule, or , states that the of the quotient of two numbers is the difference of the of the dividend and the of the divisor.
- The power rule, or , states that the of a base raised to a power is the product of the power and the of the base.
- It is best to enter expressions containing more than one into our scientific calculator as a single expression rather than calculating each individual . This helps prevent rounding errors.
