Solving a Formula For a Variable Contained in an Exponent

Learning Outcomes

• Use the power rule for logarithms to rewrite a logarithm
• Use the power rule for logarithms to solve an equation containing the variable in an exponent

Sometimes the variable of interest in an equation is contained within an exponent. For example, say we’d like to solve for $x$ in the equation $3^{x}=17$. We don’t know what number $3$ should be raised to that would result in $17$. All we know is that is is bigger than $2$ and smaller than $3$.  We’ll need to use a property of a mathematical object called a logarithm to bring the $x$ down so we can isolate it on one side of the equation. You may have studied logarithms before but, even if you didn’t, you can still use the property of taking the logarithm on both sides. Let’s build this skill up by starting with some definitions.

A LOGARITHM

logarithm is a number. Specifically, it is an exponent.

logarithm is the number $\log_{b}(M)$ to which we must raise the base $b$ in order to obtain $M$.

We call $b$ the base, and $M$ the argument of the logarithm.

When $b=10$, we call the logarithm the common logarithm and abbreviate it $\log M$.

COMMON LOGARITHM

A logarithm to base 10, $\log_{10}M$, is called the common logarithm, and is abbreviated $\log M$.

But understanding a logarithm isn’t essential to using it in the way we want to when manipulating certain formulas. Logarithms have a certain property such that, when it is applied to both sides of an equation, will bring a variable of interest down from an exponent and convert the expression into a product of the exponent and the logarithm. We call this property the power rule for logarithms.

THE POWER RULE FOR COMMON LOGARITHMS

The power rule for common logarithms, can be used to simplify the common logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.

$\log (M^{n})=n\log M$

Example

Use the power rule to write $\log\left(2^{x}\right)$ as the product of the exponent times the logarithm of the base.

Solution:

We identify the exponent, $x$, and the argument, $2^{x}$, and rewrite the equivalent expression by multiplying the exponent times the logarithm of the argument, $2$.

$\log2^{x}=x\cdot \log2$

Since $\log 2$ is a number, we can evaluate it on a calculator.

Most scientific or graphing calculators have a LOG button. Type LOG(2) and ENTER.

$\log 2 \approx 0.30103$.

So, $\log2^{x}=x\cdot \log2 \approx 0.30103x$.

This is sometimes referred to as expanding the logarithm.

Practice evaluating some common logarithms using your calculator below.

Use Common Logarithms to Solve Equations Containing Variables in the Exponents

Now that you’ve gained practice converting exponential expressions using the power rule for common logarithms and evaluating logarithms on your calculator, it’s time to learn how to apply these skills to an equation in which the variable of interest is contained in an exponent.

Example

Solve the equation for $x$.

$3^{x} = 17$

Solution:
This is the question with which we began our discussion at the top of the page. Recall that we didn’t know the exponent on 3 that would yield 17, but we knew it would be large than 2 and smaller than 3. This is because

$3^2 = 9$ and $3^3 = 27$ and $9 < 17 < 27$.

We’ll have to use a logarithm.

$3^{x} = 17$

$\log 3^{x} = \log 17$                      take the common logarithm on both sides

$x\log 3= \log 17$                      apply the power rule for common logarithm

$\dfrac{x \cancel\log 3}{\cancel\log 3}= \dfrac{\log 17}{\log 3}$                  divide $\log 3$ from both sides of the equation

$x=\dfrac{\log 17}{\log 3} \approx 2.579$                use the LOG button on a calculator to evaluate $\dfrac{\log 17}{\log 3}$ and round to 3 decimal places

Another logarithm to a special base is called the natural logarithm. This logarithm has a base of $e$, an irrational constant approximately equal to 2.718.

NATURAL LOGARITHM

A logarithm to base $e$, $\log_{e}M$, is called the natural logarithm, and is abbreviated $\ln M$.

The power rule with either the common logarithm, $\log M$, or the natural log, $\ln M$, may be used to rewrite the exponent as a product. To evaluate a natural logarithm, use the LN button on your calculator. A special characteristics of logarithms is that

$\dfrac{\log M}{\log N} = \dfrac{\ln M}{\ln N}$

The following video provides examples of using the natural logarithm or the common logarithm to solve exponential equations.

Sometimes you’ll have to do some work to isolate the term containing the exponent first before applying the power rule. See the example and video below for examples of these types of equations.