Express the given logarithm as a sum and/or difference of logarithms. Simplify, if possible. Assume that all variables represent positive real numbers.
-$\log _{19} \sqrt{\frac{m n}{3}}$
A) $\frac{1}{2} \log _{19} \mathrm{~m}+\frac{1}{2} \log _{19} \mathrm{n}-\frac{1}{2} \log _{19} 3$
B) $\frac{1}{2} \log _{19} \mathrm{mn}-\frac{1}{2} \log _{19} 3$
C) $\frac{1}{2} \log _{19} \mathrm{~m}+\frac{1}{2} \log _{19} \mathrm{n}-\log _{19} 3$
D) $\frac{1}{2} \log _{19} \mathrm{~m} \cdot \frac{1}{2} \log _{19} \mathrm{n} \div \frac{1}{2} \log _{19} 3$

Question

Quizplus · Accepted Answer

The given logarithm $\log _{19} \sqrt{\frac{m n}{3}}$ can be expressed as a sum and/or difference of logarithms by applying the logarithmic rules: the square root corresponds to a $\frac{1}{2}$ exponent, which can be brought in front of the log; the multiplication inside the log can be split into a sum of logs; and the division inside the log can be split into a difference of logs. This results in $\frac{1}{2} \log _{19} \mathrm{~m}+\frac{1}{2} \log _{19} \mathrm{n}-\frac{1}{2} \log _{19} 3$.

Express the Given Logarithm as a Sum And/or Difference of Logarithms