Chapter 3.5, Problem 12E

Solution

To find: The solution of the given equation and then check it by substituting it into the original equation.

The exact solution of the given equation ${0.98}^x=1.6$ is $x\approx -23.2644$ .

Given information:

The given equation is ${0.98}^x=1.6$ .

Concept Used:

Step involved in solving the exponential equation:

Isolate the exponential expression.

Take the common logarithm on both sides.

Apply the power rule of the logarithm, change-to-base the formula for the logarithm, and then solve.

Calculation:

Take logarithms on both sides.

  $\begin{array}{c}\log {0.98}^x=\log 1.6\\ x\log 0.98=\log 1.6\\ x=\frac{\log 1.6}{\log 0.98}\end{array}$

Use the change-to-base formula for logarithms,

  $x={\log }_{0.98}1.6$

Use the TI-83 calculator for calculating the value of ${\log }_{0.98}1.6$ .

  ${\log }_{0.98}1.6\approx -23.2644$

So,

  $x\approx -23.2644$

Now, check the result while substituting the value of $x$ in the given equation.

  ${0.98}^{-23.2644}\stackrel{?}{=}1.6$

Use the TI-83 calculator for calculating the value of ${0.98}^{-23.2644}$ .

  $\begin{array}{c}{0.98}^{-23.2644}\approx 1.6\\ 1.6=1.6\left(\text{True}\right)\end{array}$

Hence, the value of $x$ is $\approx -23.2644$ .