Answer
$t=4$
Work Step by Step
Using the properties of logarithms, the given equation, $
\log_2(t+5)-\log_2 (t-1)=\log_2 3
$, is equivalent to
\begin{align*}\require{cancel}
\log_2\dfrac{t+5}{t-1}&=\log_2 3
&(\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{t+5}{t-1}&=3
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
(\cancel{t-1})\cdot\dfrac{t+5}{\cancel{t-1}}&=3\cdot(t-1)
\\\\
t+5&=3t-3
\\
5+3&=3t-t
\\
8&=2t
\\\\
\dfrac{8}{2}&=\dfrac{\cancel2t}{\cancel2}
\\\\
4&=t
\\
t&=4
.\end{align*}
Hence, the solution to the equation $
\log_2(t+5)-\log_2 (t-1)=\log_2 3
$ is $
t=4
$.