Use properties of logarithms to write each expression as a sum or difference of logarithms. Assume that variables represent positive real numbers.
-$\log _{7} \frac{\sqrt[9]{5}}{n^{2} m}$
A) $\frac{1}{9} \log _{7} 5-2 \log 7 n-2 \log 7 m$
B) $9 \log _{7} 5-2 \log 7 n-\log _{7} 8$
C) $\frac{1}{9} \log _{7} 5-2 \log 7 n-\log _{7} \mathrm{~m}$
D) $\log _{7} 5-\log 7 n-\log 7 m$

Question

Quizplus · Accepted Answer

Using the properties of logarithms, the given expression $\log _{7} \frac{\sqrt[9]{5}}{n^{2} m}$ can be rewritten as a sum or difference of logarithms. The properties used include the quotient rule $\log_b \frac{x}{y} = \log_b x - \log_b y$, the power rule $\log_b x^y = y \log_b x$, and the root rule which is a specific case of the power rule $\log_b \sqrt[n]{x} = \frac{1}{n} \log_b x$. Applying these rules, the expression simplifies to $\frac{1}{9} \log _{7} 5-2 \log _{7} n-\log _{7} m$.

Use Properties of Logarithms to Write Each Expression as a Sum