Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
-$\log _ { 7 } \left( \frac { \sqrt [ 3 ] { p } \sqrt [ 5 ] { q } } { t ^ { 2 } } \right)$
A) $\frac { 1 } { 3 } \log _ { 7 } p + \frac { 1 } { 5 } \log _ { 7 } q - 2 \log _ { 7 } t$
B) $\frac { 1 } { 3 } \log _ { 7 } p \cdot \frac { 1 } { 5 } \log _ { 7 } q \div 2 \log _ { 7 } \mathrm { t }$
C) $\frac { 3 } { 7 } \log _ { 7 } p + \frac { 5 } { 7 } \log _ { 7 } q - \frac { 2 } { 7 } \log _ { 7 } \mathrm { t }$
D) $3 \log _ { 7 } p + 5 \log _ { 7 } q - 2 \log _ { 7 } \mathrm { t }$

Question

Quizplus · Accepted Answer

The expression can be expanded using the properties of logarithms: the quotient rule $\log_b \frac{M}{N} = \log_b M - \log_b N$, the product rule $\log_b (MN) = \log_b M + \log_b N$, and the power rule $\log_b M^k = k \log_b M$. Applying these rules, we get $\frac{1}{3}\log_7 p + \frac{1}{5}\log_7 q - 2\log_7 t$.

Use Properties of Logarithms to Expand the Logarithmic Expression as Much