Basic Technical Mathematics, 11th Edition

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

1. 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.

2. 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 ...