Chapter 6.5, Problem 2E

To determine

To calculate: The ways to evaluate the logarithmic expression ${\log }_{7}12$ using a calculator.

Answer to Problem 2E

The two ways are change of base formula and product property of logarithm.

Explanation of Solution

Given information:

The given expression is ${\log }_{7}12$ .

Formulas used:

Change of base formula:

  ${\log }_{n}m=\frac{\log m}{\log n}$

Product property of logarithm:

  $\log ab=\log a+\log b$

Calculation:

Consider the expression ${\log }_{7}12$ .

Rewrite the expression by, ${\log }_{n}m=\frac{\log m}{\log n}$ .

Substitute $1.08$ for $\log 12$ and $0.84$ for $\log 7$ and solve the expression by using a calculator.

  $\begin{array}{c}{\log }_{7}12=\frac{\log 12}{\log 7}\\ =\frac{1.08}{0.84}\\ =1.27\end{array}$

Rewrite the expression by using the product property of logarithm $\log ab=\log a+\log b$

  $\begin{array}{c}{\log }_{7}12=\log \left(4\times 3\right)\\ ={\log }_{7}4+{\log }_{7}3\end{array}$

Use the change of base formula to simplify the above expression, ${\log }_{n}m=\frac{\log m}{\log n}$ .

  $\begin{array}{c}{\log }_{7}12={\log }_{7}4+{\log }_{7}3\\ =\frac{\log 4}{\log 7}+\frac{\log 3}{\log 7}\end{array}$

Substitute $0.60$ for $\log 4$ , $0.48$ for $\log 3$ and $0.84$ for $\log 7$ and solve the expression by using a calculator.

  $\begin{array}{c}{\log }_{7}12=\frac{\log 4}{\log 7}+\frac{\log 3}{\log 7}\\ =\frac{0.60}{0.84}+\frac{0.48}{0.84}\\ =1.27\end{array}$

Hence, the two ways are change of base formula and product property of logarithm.