Answer
$x=3$
Work Step by Step
Using the properties of logarithms, the given equation, $
\log (-x)+\log3=\log(2x-15)
$, is equivalent to
\begin{align*}\require{cancel}
\log (-x\cdot3)&=\log(2x-15)
&(\text{use }\log_b (xy)=\log_b x+\log_b y)
\\
\log (-3x)&=\log(2x-15)
.\end{align*}
Since $\log_b m=\log_b n $ implies $m=n$, the equation above implies
\begin{align*}\require{cancel}
-3x&=2x-15
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
15&=2x+3x
\\
15&=5x
\\\\
\dfrac{15}{5}&=\dfrac{\cancel5x}{\cancel5}
\\\\
3&=x
\\
x&=3
.\end{align*}
Hence, the solution to the equation $
\log (-x)+\log3=\log(2x-15)
$ is $
x=3
$.
