Does it matter which base logarithm I use?

Usually, no. You can take the natural log ($\ln$) or the common log ($\log$) of both sides, and as long as you do the same thing to both sides, you will get the correct answer. However, if your equation has a base of $e$, it is highly recommended to use $\ln$, and if it has a base of 10, use $\log$, as this makes the math much simpler.

What if there are variables in the exponents on both sides?

The process is exactly the same. Take the logarithm of both sides, then use the power rule to bring both exponents down. You will then distribute, group all the terms with an $x$ on one side of the equation, factor out the $x$, and divide to solve.

Can I solve an exponential equation without a calculator?

You can find the exact algebraic answer, like $x = \frac{\ln(7)}{2}$. This is considered the "exact form" and is perfectly acceptable (and often preferred) in math classes. To find the decimal approximation, however, you will need a calculator.

How do you solve an equation where the variable is in the exponent?

To solve an equation where the variable is in the exponent, you use logarithms. A logarithm is the mathematical inverse of an exponent. By applying a logarithm to both sides of the equation, you can pull the variable down from the exponent so it becomes a regular multiplication problem.

Think of the variable in the exponent like a cat stuck high up in a tree. You cannot reach it to do your normal algebraic steps. The logarithm is the ladder that brings the cat safely down to the ground. Once the variable is on the ground level, you can solve for it just like you would in any standard algebra problem.

What is an exponential equation?

An exponential equation is simply any equation where the variable we want to find is located in the exponent. For example, in the equation $2^x = 8$ , the variable $x$ is the exponent. While you might be able to guess the answer to $2^x = 8$ just by knowing that $2^3 = 8$ , most equations, like $5^x = 12$ , do not have neat whole-number answers. You need a reliable mathematical tool to solve them.

Why logarithms work

Logarithms work because of a special property called the power rule of logarithms. The rule states that $\log(a^x) = x \log(a)$ . Notice what happened to the $x$ . It moved from being an exponent to being a coefficient multiplied by the logarithm. This is the "ladder" in our analogy. Because a logarithm is the exact opposite of an exponential function, taking the log of an exponential function "undoes" the base, leaving you with a standard algebraic equation.

How to solve step-by-step

First, you must isolate the exponential part of the equation on one side. If your equation is $3(10^x) = 15$ , divide by 3 first to get $10^x = 5$ . Second, take the logarithm of both sides. You can use the common logarithm (base 10, written as $\log$ ) or the natural logarithm (base $e$ , written as $\ln$ ). For $10^x = 5$ , taking the common log gives $\log(10^x) = \log(5)$ . Third, use the power rule to bring the variable down: $x \log(10) = \log(5)$ . Finally, solve for $x$ . Since $\log(10) = 1$ , we get $x = \log(5)$ .

Where students slip up

The most common mistake students make is applying the logarithm before isolating the base. If you have $4 \cdot 2^x = 20$ and try to take the log immediately, you will get stuck because $\log(4 \cdot 2^x)$ does not simplify easily. Always divide by that 4 first to get $2^x = 5$ . Another common slip is trying to take the logarithm of a negative number. If your isolation step leaves you with $e^x = -4$ , you can stop there: an exponential function with a positive base can never equal a negative number, so there is no real solution.

Worked through

Solve for $x$ : $3 e^{2x} - 5 = 16$

Step 1: Isolate the exponential part. Add 5 to both sides. $3 e^{2x} = 21$

Step 2: Divide both sides by 3. $e^{2x} = 7$

Step 3: Take the natural logarithm ( $\ln$ ) of both sides, because the base is $e$ . $\ln(e^{2x}) = \ln(7)$

Step 4: Use the power rule to bring down the exponent. (Remember that $\ln(e) = 1$ ). $2x = \ln(7)$

Step 5: Divide by 2 to solve for $x$ . $x = \frac{\ln(7)}{2}$

Questions students ask

Ask about this topic

Explain at

Where this comes from: OpenStax College Algebra, Chapter 6: Exponential and Logarithmic Functions · Khan Academy, Algebra 2: Solving exponential equations with logarithms