#### Answer

{$\frac{1}{9}$}

#### Work Step by Step

Step 1: $\log (x+1)-\log x=1$
Step 2: $\log \frac{x+1}{x}=1$
Step 3: $\frac{x+1}{x}=10^{1}$
Step 4: $\frac{x+1}{x}=10$
Step 5: $10x=x+1$
Step 6: $9x=1$
$x=\frac{1}{9}$
Therefore, the solution set is {$\frac{1}{9}$}.