# 13.6 Exponential and Logarithmic Equations

**Exponential Equations • Logarithmic Equations**

# EXPONENTIAL EQUATIONS

*An equation in which the variable occurs in an exponent is called an* **exponential equation.** Although some may be solved by changing to logarithmic form, they are generally solved by *taking the logarithm of each side* and then using the properties of logarithms.

# EXAMPLE 1 Exponential equation—solved two ways

We can solve the exponential equation ${2}^{x}\hspace{0.17em}=\hspace{0.17em}8$ by writing it in logarithmic form. This gives

$$\begin{array}{cc}x\hspace{0.17em}=\hspace{0.17em}{\hspace{0.17em}log\hspace{0.17em}}_{2}8\hspace{0.17em}=\hspace{0.17em}3& {2}^{3}\hspace{0.17em}=\hspace{0.17em}8\end{array}$$This method is good if we can directly evaluate the resulting logarithm.

Because ${2}^{x}$ and 8 are equal, the logarithms of ${2}^{x}$ and 8 are also equal. Therefore, we can also solve ${2}^{x}\hspace{0.17em}=\hspace{0.17em}8$ in a more ...

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