Expand. Assume that all variables represent positive real numbers.
-$\log _ { 11 } \left( \frac { 5 \sqrt { m } } { n } \right)$
A) $\log _ { 11 } 5 \cdot \frac { 1 } { 2 } \log _ { 11 } \mathrm {~m} \div \log _ { 11 } \mathrm { n }$
B) $\log _ { 11 } ( 5 \sqrt { \mathrm { m } } ) - \log _ { 11 } \mathrm { n }$
C) $\log _ { 11 } n - \log _ { 11 } 5 - \frac { 1 } { 2 } \log _ { 11 } m$
D) $\log _ { 11 } 5 + \frac { 1 } { 2 } \log _ { 11 } \mathrm {~m} - \log _ { 11 } \mathrm { n }$

Question

Quizplus · Accepted Answer

The correct expansion uses the logarithm properties: $\log_b(xy) = \log_b x + \log_b y$ and $\log_b(x^y) = y\log_b x$. The division inside the logarithm is handled as subtraction: $\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y$. Therefore, $\log_{11}\left(\frac{5\sqrt{m}}{n}\right)$ expands to $\log_{11}5 + \frac{1}{2}\log_{11}m - \log_{11}n$.

Expand Assume That All Variables Represent Positive Real Numbers log⁡11(5mn)\log _ { 11 } \left( \frac { 5 \sqrt { m } } { n } \right)log11​(n5m​​)

Expand Assume That All Variables Represent Positive Real Numbers $\log _ { 11 } \left( \frac { 5 \sqrt { m } } { n } \right)$