Chapter 4, Problem 9CRT

a.

To determine

To find: the exact value of the expression.

  $\log 16-2{\log }_{3}36$

Answer to Problem 9CRT

The value of the expression is -4.

Explanation of Solution

Given information:

The expression is given.

Formula used:

Properties of logarithmic function

  $\begin{array}{l}{\log }_{c}a-\log b=\log \frac{a}{b}\\ a\log b=\log b^a\\ \log a=1\end{array}$

Calculation:

Consider that expression is $\log 16-2{\log }_{3}36$

Recall that the formula is

  $\begin{array}{l}{\log }_{c}a-\log b=\log \frac{a}{b}\\ a\log b=\log b^a\\ \log a=1\end{array}$

Therefore

  $\begin{array}{l}\log 16-2\log 36\\ \log 16-\log 36^2\\ \log 16-\log 1296\\ \log \frac{16}{1296}\\ \log \frac{1}{81}\\ $ \log (1)-\log (81)$$ 0-\log (3){}^4$-4\log 3\\ -4\end{array}$

So, the exact values of the given expression is -4

b.

To determine

To find: the expression for $\log (\frac{x^5\sqrt{x-1}}{2x-3})$

Answer to Problem 9CRT

The expand form of the expression is $5\log x+\frac{1}{2}\log (x-1)-\log (2x-3)$

Explanation of Solution

Given information:

The expression is given.

Formula used : by the properties of logarithm

  $\begin{array}{l}{\log }_{c}a-\log b=\log \frac{a}{b}\\ a\log b=\log b^a\\ {\log }_{c}a+\log b=\log (ab)\end{array}$

Calculation:

Consider that expression is

  $\log (\frac{x^5\sqrt{x-1}}{2x-3})$

Recall that the formula of logarithm is $\begin{array}{l}{\log }_{c}a-\log b=\log \frac{a}{b}\\ a\log b=\log b^a\\ {\log }_{c}a+\log b=\log (ab)\end{array}$

Therefore , $\log (\frac{x^5\sqrt{x-1}}{2x-3})$

  $\begin{array}{l}\log (x^5\sqrt{x-1})-\log (2x-3)\\ \log x^5+\log \sqrt{x-1}-\log (2x-3)\\ 5\log x+\log {(}^x-\log (2x-3)\\ 5\log x+\frac{1}{2}\log (x-1)-\log (2x-3)\end{array}$

So the expand form of the expression is $5\log x+\frac{1}{2}\log (x-1)-\log (2x-3)$ .