7.6 Solving Exponential Equations

Learning Outcomes

  • Solve exponential equations with common bases.

When an exponential equation has the same base on each side, the exponents must be equal. This is a consequence of the One-to-One Property of Exponents, which previously allowed us to define its inverse function, the logarithm.

An example of such an equation is [latex]2^x=2^5[/latex]. Since the exponential function is one-to-one, the exponent on the left must be [latex]5[/latex] in order for the two sides to equal. No other exponent will result in the same value on both sides.

ONE-TO-ONE PROPERTY OF EXPONENTS

Let [latex]b>0[/latex] and [latex]b \neq 1.[/latex] Then [latex]b^x = b^y[/latex] implies [latex]x=y.[/latex]

This also applies when the exponents are algebraic expressions.

Therefore, we can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then, we can set the exponents equal to one another and solve for the unknown.

Consider the equation [latex]{3}^{4x - 7}=3^{2x-1}[/latex]. To solve for [latex]x[/latex], we apply the one-to-one property of exponents by setting the exponents equal to one another:

[latex]\begin{align}3^{4x - 7} &=3^{2x - 1} \\ 4x - 7 &=2x - 1&& \color{blue}{\textsf{apply the One-to-One Property of Exponents}} \\ 2x &=6 && \color{blue}{\textsf{subtract 2}x\textsf{ and add 7 to both sides}}\\ x& =3 && \color{blue}{\textsf{divide by 2}}\end{align}[/latex]

In our first example, we solve an exponential equation whose terms all have a common base.

Example

Solve [latex]{2}^{x - 1}={2}^{2x - 4}[/latex].

Rewriting Equations So All Powers Have the Same Base

Sometimes if the bases of an exponential equation are not equal, we can rewrite the terms as powers with a common base and solve using the One-to-One Property of Exponents. For example, you can rewrite 8 as [latex]2^3[/latex] or 36 as [latex]6^2[/latex] or [latex]\frac{1}{4}[/latex] as [latex]\left(\frac{1}{2}\right)^{2}[/latex] or [latex]2^{-2}.[/latex]

Consider the equation [latex]256={4}^{x - 5}[/latex]. We can rewrite both sides of this equation as a power of [latex]2[/latex]. Then we apply the rules of exponents, along with the One-to-One Property of Exponents, to solve for [latex]x[/latex]:

[latex]\begin{align}256&={4}^{x - 5} \\ 2^8&=\left(2^2\right)^{x-5} && \color{blue}{\textsf{rewrite the base on each side as a power of }2} \\ 2^8&=2^{2\cdot(x-5)} && \color{blue}{\textsf{use the Power Rule for Exponents}} \\ 8&=2x-10 && \color{blue}{\textsf{apply the One-to-One Property of Exponents}} \\ 18&=2x \\ 9&=x \end{align}[/latex]

The next example is similar but both exponents contain variables.

Example

Solve [latex]{8}^{x+2}={16}^{x+1}[/latex].

Remember that you can write radicals as rational exponents, so you may be able to find common bases when it is not completely obvious at first.

Example

Solve [latex]{5}^{7x}=\sqrt{5}[/latex].

In the following video, we show more examples of how to solve exponential equations by finding a common base.

Before revealing the answer to the example, think about the range of the exponential function, which is the set of output values it is allowed to produce.

Example

Solve [latex]{3}^{x+1}=-2[/latex].

 Analysis of the Solution

The figure below shows the graphs of the two separate expressions in the equation [latex]{3}^{x+1}=-2[/latex] as [latex]y={3}^{x+1}[/latex] and [latex]y=-2[/latex]. The two graphs do not cross since the exponential function has a horizontal asymptote of [latex]y=0,[/latex] showing us that the left side is never equal to the right side. Thus the equation has no solution.

Graph of 3^(x+1)=-2 and y=-2. The graph notes that they do not cross.

Summary

We can use the One-to-One Property of Exponents to solve exponential equations whose bases are the same by setting the exponents equal to each other. The terms in some exponential equations can be rewritten with the same base, allowing us to use the same principle. There are exponential equations that do not have solutions because the exponential function can only produce positive values as outputs.