Answer
No solution.
Work Step by Step
First, the solutions must satisfy $\left\{\begin{array}{l}
x+1\gt 0\\
x\gt 0
\end{array}\right.\quad \Rightarrow x\gt 0\qquad (*)$
in order for the equation to be defined.
On the LHS, apply $\quad\displaystyle \log_{a}\frac{M}{N}=\log_{a}M-\log_{a}N$
On the RHS, apply$\quad \log 10^{0}=0$
$\displaystyle \log(\frac{x+1}{x})=\log 1$
... apply the principle of logarithmic equality
$\displaystyle \frac{x+1}{x}=1\qquad $... multiply with $x$
$x+1=x\qquad $ ... subtract x
$1=0$
No solution.
