Chapter 3.3, Problem 30E

Solution

To evaluate: the given logarithmic expression if it is defined using a calculator, and check your result by evaluating the corresponding exponential expression.

  $\ln 0.733\approx -0.311$

Given information:

The expression:

  $\ln 0.733$

Definition Used:

Natural logarithm- Base e :

The Natural logarithmic function ${\log }_{e}x=\ln x$ is the inverse of the exponential function $f\left(x\right)=e^x$ . Therefore,

  $y=\ln x$ if and only if $e^y=x$

Explanation:

First use a calculator to evaluate $\ln 0.733$ .

  

So,

  $\begin{array}{c}\ln 0.733=-0.3106095771\\ \approx -0.311\end{array}$

Now, evaluating the corresponding exponential expression.

By the above definition of natural logarithm:

  $y=\ln x$ if and only if $e^y=x$

Applying this to the given function:

  $\ln 0.733=-0.3106095771$ if and only if $e^{-0.3106095771}=0.733$

So, checking this using a calculator,