Chapter 4.FR, Problem 1E

Reminder Round all answers to two decimal places unless otherwise indicated.

Using the Laws of Logarithms For Exercises 1 through 6, suppose that $\ln A=3,$ $\ln B=4$, and $\ln C=5$. Evaluate the given expression.

$\ln \frac{AB}{C}$

To determine

To evaluate:

The given expression.

Answer to Problem 1E

Solution:

The expression $\ln \frac{AB}{C}$ is equal to $2$.

Explanation of Solution

Given:

The values $\ln A=3,$ $\ln B=4$, and $\ln C=5$.

The expression is $\ln \frac{AB}{C}$.

Approach:

Following laws of logarithms are to be referred wherever applicable:

1) Product law: $\ln AB=\ln A+\ln B$

2) Quotient law: $\ln \frac{A}{B}=\ln A-\ln B$

3) Power law: $\ln A^p=p\ln A$

Calculation:

Consider the expression.

$\begin{array}{rll}\ln \frac{AB}{C}& =& \ln AB-\ln C\\ & =& \ln A+\ln B-\ln C\end{array}$

Substitute $\ln A=3,$ $\ln B=4$, and $\ln C=5$ in the above equation.

$\begin{array}{rll}\ln A+\ln B-\ln C& =& 3+4-5\\ & =& 7-5\\ & =& 2\end{array}$

Therefore, the expression $\ln \frac{AB}{C}$ is equal to $2$.

Conclusion:

Hence, the expression $\ln \frac{AB}{C}$ is equal to $2$.