Chapter 3.5, Problem 11E

Solution

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

The exact solution of the given equation ${1.06}^x=4.1$ is $x\approx 24.2151$ .

Given information:

The given equation is ${1.06}^x=4.1$ .

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 logarithm on both sides.

  $\begin{array}{c}\log {1.06}^x=\log 4.1\\ x\log 1.06=\log 4.1\\ x=\frac{\log 4.1}{\log 1.06}\end{array}$

Use the change-to-base formula for logarithms,

  $x={\log }_{1.06}4.1$

Use the TI-83 calculator for calculating the value of ${\log }_{1.06}4.1$ .

  ${\log }_{1.06}4.1\approx 24.2151$

So,

  $x\approx 24.2151$

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

  ${1.06}^{24.2151}\stackrel{?}{=}4.1$

Use the TI-83 calculator for calculating the value of ${1.06}^{24.2151}$ .

  $\begin{array}{c}{1.06}^{24.2151}\approx 1.6\\ 1.6=1.6\left(\text{True}\right)\end{array}$

Hence, the value of $x$ is $\approx 24.2151$ .