Answer
$\log_3{u}+2\log_3{v}-\log_3{w}$
Work Step by Step
We wish to find $\log_3{u}+\log_3{v^2}-\log_3{w}$. In order to solve this problem, we will use the following logarithmic properties:
(a) $\sqrt[m]{a}=a^{\frac{1}{m}}$
(b) $\log_a {x^n}=n\cdot \log_a {x}$
(c) $\log_a{xy}=\log_a{x} +\log_a{y}$
(d) $\log_a{\dfrac{x}{y}}=\log_a{x} -\log_a{y}$
($\log_a{M}=\log_a{N} \longrightarrow M=N$.)
Use property: $\log_a{\dfrac{x}{y}}=\log_a{x} -\log_a{y}$
Thus, $\log_3{\frac{uv^2}{w}}=\log_3{uv^2}-\log_3{w}.$
Use property: $\log_a{\dfrac{x}{y}}=\log_a{x} -\log_a{y}$
So, $\log_3{uv^2}-\log_3{w}=\log_3{u}+\log_3{v^2}-\log_3{w}$
$\log_a {x^n}=n\cdot \log_a {x}$
So, $\log_3{u}+\log_3{v^2}-\log_3{w}=\log_3{u}+2\log_3{v}-\log_3{w}$
