Expand into a sum and/or difference of logarithms of a and b. Assume all variables represent positive real numbers. $\log _ { 8 } \left( \frac { a ^ { 4 } } { \sqrt { b } } \right)$
A) $4 \log _ { 8 } a - \frac { 1 } { 2 } \log _ { 8 } b$
B) $\frac { \log _ { 8 } a ^ { 4 } } { \frac { 1 } { 2 } \log _ { 8 } b }$
C) $\log _ { 8 } 4 a + \log _ { 8 } \frac { b } { 2 }$
D) $4 \log _ { 8 } a + \frac { 1 } { 2 } \log _ { 8 } b$

Question

Quizplus · Accepted Answer

Using the properties of logarithms, the expression $\log _ { 8 } \left( \frac { a ^ { 4 } } { \sqrt { b } } \right)$ can be expanded as $4 \log _ { 8 } a - \frac { 1 } { 2 } \log _ { 8 } b$. This uses the rule that $\log(a^m) = m\log(a)$ and $\log\left(\frac{a}{b}\right) = \log(a) - \log(b)$.

Expand into a Sum And/or Difference of Logarithms of a and B