Chapter 8.5, Problem 58PPS

To determine

To find: Whether the equation is True or False.

Answer to Problem 58PPS

The above equation is equal so the given equation is True.

Explanation of Solution

Given:

The given equation is ${\log }_9\frac{x^2{y}^3}{z^4}=2{\log }_{9}x+3{\log }_{9}y-4{\log }_{9}z$ .

Calculation:

Consider the given equation is,

  ${\log }_9\frac{x^2{y}^3}{z^4}=2{\log }_{9}x+3{\log }_{9}y-4{\log }_{9}z$

Solve the above equation as,

  $\begin{array}{c}{\log }_9\frac{x^2{y}^3}{z^4}=2{\log }_{9}x+3{\log }_{9}y-4{\log }_{9}z\\ ={\log }_9{x}^2+{\log }_9{y}^3-{\log }_9{z}^4\\ ={\log }_9\left(x^2{y}^3\right)-{\log }_9{z}^4\\ ={\log }_9\left(\frac{x^2{y}^3}{z^4}\right)\end{array}$

Since, the above equation is equal so the given equation is True.