Answer
$t=2$
Work Step by Step
Using the properties of logarithms, the given equation, $
\log_5(3t+2)-\log_5 t=\log_5 4
$, is equivalent to
\begin{align*}\require{cancel}
\log_5\dfrac{3t+2}{t}&=\log_5 4
&(\text{use }\log_b \dfrac{x}{y}=\log_b x-\log_b y)
.\end{align*}
Since $\log_b m=\log_b n $ implies $m=n$, the equation above implies
\begin{align*}\require{cancel}
\dfrac{3t+2}{t}&=4
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
\cancel t\cdot\dfrac{3t+2}{\cancel t}&=4\cdot t
\\
3t+2&=4t
\\
2&=4t-3t
\\
2&=t
\\
t&=2
.\end{align*}
Hence, the solution to the equation $
\log_5(3t+2)-\log_5 t=\log_5 4
$ is $
t=2
$.
