## Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole

# Chapter 4 - Section 4.4 - Laws of Logarithms - 4.4 exercises: 57

#### Answer

$\log (\frac{x^2}{(x-3)})$

#### Work Step by Step

$Combine$ $the$ $expression$: $\frac{1}{3}$$\log (x+2)^3 + \frac{1}{2}$$[$$\log x^4 - \log (x^2 - x - 6)^2$$]$ Distribute the half to the variables in the brackets $\frac{1}{3}$$\log (x+2)^3 + \frac{1}{2}$$\log x^4$ - $\frac{1}{2}$$\log (x^2 - x - 6)^2 Apply the Third Law of Logarithms for all the terms \frac{1}{3}$$\log (x+2)^3$ = $\log (x+2)^{3\times \frac{1}{3}}$ $\frac{1}{2}$$\log x^4 = \log x^{4\times \frac{1}{2}} \frac{1}{2}$$\log (x^2 - x - 6)^2$ = $\log (x^2-x-6)^{2\times \frac{1}{2}}$ $\log (x+2)$ + $\log x^2$ - $\log (x^2-x-6)$ Apply the First Law of Logarithms for $\log (x+2)$ + $\log x^2$ $\log (x+2)$ + $\log x^2$ = $\log ((x+2)\times (x^2))$ $\log (x^2(x+2))$ - $\log (x^2-x-6)$ Apply the Second Law of Logarithms $\log (x^2(x+2))$ - $\log (x^2-x-6)$ = $\log (\frac{x^2(x+2)}{x^2-x-6})$ Factor x$^2$-x-6 $\log (\frac{x^2(x+2)}{(x+2)(x-3)})$ Simplify $\log (\frac{x^2}{(x-3)})$

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