Answer
$\left\{1\right\}$
Work Step by Step
Using the properties of logarithms, the given equation, $
\log_2 x+\log_2 (x+15)=\log_2 16
$, is equivalent to
\begin{align*}\require{cancel}
\log_2 [x(x+15)]&=\log_2 16
&(\text{use }\log_b (xy)=\log_b x+\log_b y)
\\
\log_2 (x^2+15x)&=\log_2 16
&(\text{use the Distributive Property})
.\end{align*}
Since $\log_b x=\log_b y$ implies $x=y$, the equation above implies
\begin{align*}\require{cancel}
x^2+15x&=16
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
x^2+15x-16&=0
.\end{align*}
Using the factoring of trinomials, the equation above is equivalent to
\begin{align*}\require{cancel}
(x+16)(x-1)&=0
.\end{align*}
Equating each factor to zero (Zero Product Property) and solving for the variable, then
\begin{array}{l|r}
x+16=0 & x-1=0
\\
x=-16 & x=1
.\end{array}
If $x=-16$, the term $\log_2x$ of the original equation becomes $\log_2(-16)$. This is undefined since in $\log_b x$, $x$ and $b$ should be positive numbers with $b\ne1$.
Hence, the solution set of the equation $
\log_2 x+\log_2 (x+15)=\log_2 16
$ is $
\left\{1\right\}
$.
