Chapter 13.1, Problem 56E

To determine

To calculate: The value of x in the expression ${7.1}^x=83.1$ .

Answer to Problem 56E

The value of x in the expression ${7.1}^x=83.1$ is $2.26$ .

Explanation of Solution

Given information:

The expression ${7.1}^x=83.1$ .

Formula used:

The property of logarithmic function is $\log a^x=x\log a$ where $a>0$ .

Calculation:

Consider the expression ${7.1}^x=83.1$ .

Take logarithm on both the sides,

  $\log \left({7.1}^x\right)=\log \left(83.1\right)$

Apply the property of logarithmic function that is $\log a^x=x\log a$ where $a>0$ .

  $\begin{array}{c}\log \left({7.1}^x\right)=\log \left(83.1\right)\\ x\log \left(7.1\right)=\log \left(83.1\right)\\ x=\frac{\log \left(83.1\right)}{\log \left(7.1\right)}\end{array}$

Use a scientific calculator to simplify the above expression,

  $\begin{array}{c}x=\frac{\log \left(83.1\right)}{\log \left(7.1\right)}\\ =\frac{4.4200}{1.9601}\\ =2.2550\\ =2.26\end{array}$

Thus, the value of x in the expression ${7.1}^x=83.1$ is $2.26$ .