Chapter 6.5, Problem 15E

To determine

To calculate: The expanded form of the expression $\log 10x^5$ .

Answer to Problem 15E

The required expanded form of the expression $\log 10x^5$ is $1+5\log x$ .

Explanation of Solution

Given information:

The given expression is $\log 10x^5$ .

Formula used:

Product property of logarithm:

  ${\log }_{b}m+{\log }_{b}n={\log }_{b}mn$

Power property of logarithm:

  ${\log }_b{m}^n=n{\log }_{b}m$ .

Calculation:

Consider the expression $\log 10x^5$ .

Expand the expression using product property of logarithm ${\log }_{b}mn={\log }_{b}m+{\log }_{b}n$ .

  $\log 10x^5=\log 10+\log x^5$

Expand the above expression using power property of logarithm ${\log }_b{m}^n=n{\log }_{b}m$ .

  $\begin{array}{c}\log 10x^5=\log 10+\log x^5\\ =\log 10+5\log x\end{array}$

Substitute $1$ for $\log 10$ in the above expression.

  $\begin{array}{c}\log 10x^5=\log 10+5\log x\\ =1+5\log x\end{array}$

Hence, the required expanded form of the expression ${\log }_{8}3x$ is $1+5\log x$ .