[tex]\bf \textit{Logarithm of rationals}\\\\
log_{{ a}}\left( \frac{x}{y}\right)\implies log_{{ a}}(x)-log_{{ a}}(y)
\\\\\\
\textit{Logarithm of exponentials}\\\\
log_{{ a}}\left( x^{{ b}} \right)\implies {{ b}}\cdot log_{{ a}}(x)\\\\\\
\textit{also recall that }\qquad a^{\frac{{ n}}{{ m}}} \implies \sqrt[{ m}]{a^{ n}} \qquad \textit{and that }\qquad log_x(x)=1\\\\
-------------------------------\\\\[/tex]
[tex]\bf log_5\left( \cfrac{125}{\sqrt{x+6}} \right)\qquad
\begin{cases}
125=5^3\\\\
\sqrt{x+6}=(x+6)^{\frac{1}{2}}
\end{cases}\implies log_5\left[ \cfrac{5^3}{(x+6)^{\frac{1}{2}}} \right]
\\\\\\
log_5(5^3)-log_5\left[ (x+6)^{\frac{1}{2}} \right]\implies 3log_5(5)-\cfrac{1}{2}log_5(x+6)
\\\\\\
3(1)-\cfrac{1}{2}log_5(x+6)\implies 3-\cfrac{log_5(x+6)}{2}[/tex]
