Chapter 3.3, Problem 36E

To determine

To verify: The given equation.

Explanation of Solution

Given:

The following equation:

  $\ln \left(\frac{98}{e}\right)=2\ln 7+\ln 2-1$

Formula used:

The following quotient property:

  $\ln \left(\frac{u}{v}\right)=\ln u-\ln v$

The following product property:

  $\ln \left(uv\right)=\ln u+\ln v$

The following power property:

  $\ln u^n=n\ln u$

The following property of natural logarithms:

  $\ln e=1$

Calculation:

The given equation can be verified as follows:

  $\begin{array}{l}\ln \left(\frac{98}{e}\right)=\ln 98-\ln e[\text{Quotient property]}\\ =\ln 98-1[\ln e=1]\\ =\ln \left(7^2\cdot 2\right)-1[{\text{Rewrite 98 as 7}}^2\cdot 2]\\ =\ln 7^2+\ln 2-1[\text{Product property]}\\ =2\ln 7+\ln 2-1[\text{Power property]}\end{array}$

Since the left hand side of the equation is equal to the right hand side, the equation is verified.