Write the expression in terms of the logarithms of a, b, and c. ln((sqrt(ab^8))/c)

Question

anonymous · Answer

$$\begin{align}
\ln\left(\frac{\sqrt{ab^8}}{c}\right)&=\ln(\sqrt{ab^8})-\ln(c) \
&=\ln[(ab^8)^{1/2}]-\ln(c) \
&=\frac{1}{2}\ln(ab^8)-\ln(c) \
&=\frac{1}{2}[\ln(a)+\ln(b^8)]-\ln(c) \
&=\frac{1}{2}[\ln(a)+8\ln(b)]-\ln(c) \
&=\frac{1}{2}\ln(a)+4\ln(b)-\ln(c)
\end{align}$$

anonymous · Answer

These question usually involve the applicaiton of the three logarithm rules (as $\ln(x)$ is just another logarithm!):
$$\log_z\left(\frac{x}{y}\right)=\log_z(x)-\log_z(y) \
\log_z\left({x}{y}\right)=\log_z(x)+\log_z(y) \
\log_zx^y=y\log_zx$$