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