Algebra and Trigonometry, 5th Edition

5.5 Solving Exponential Equations and Logarithmic Equations

• Solve exponential equations.

• Solve logarithmic equations.

Solving Exponential Equations

Equations with variables in the exponents, such as

$$3^x=20\mathrm{and}{2}^{5x}=64,$$

are called exponential equations.

Sometimes, as is the case with the equation $2^{5x}=64$, we can write each side as a power of the same number:

$$2^{5x}=2^6.$$

We can then set the exponents equal and solve:

$$\begin{array}{rcl}5x& =& 6\\ x& =& \frac{6}{5},\mathrm{or}\mathrm{1.2.}\end{array}$$

We use the following property to solve exponential equations.

Base–Exponent Property

For any $a>0$, $a\ne 1$,

$$a^x=a^y\leftrightarrow x=y.$$

This property follows from the fact that for any $a>0$, $a\ne 1$, $f(x)=a^x$ is a one-to-one function. If $a^x=a^y$, then $f(x)=f(y)$. Then since f is one-to-one, it follows that $x=y$. Conversely, if $x=y$, it follows ...