Answer
No solution or $\varnothing $.
Work Step by Step
The given equation is
$\log (2x-1) -\log (x)=2$
Use quotient rule.
$\log \left [\frac{2x-1}{x} \right]=2$
Rewrite the common logarithm showing base $10$.
$\log_{10} \left [\frac{2x-1}{x} \right]=2$
Rewrite in exponential form.
$\frac{2x-1}{x}= 10^2$
Multiply both sides by $x$.
$(x)\cdot \frac{2x-1}{x}= (x)\cdot 100$
Clear the parentheses.
$2x-1=100x$
Add $-2x$ to both sides.
$2x-1-2x=100x-2x$
Simplify.
$-1=98x$
Divide both sides by $98$.
$-\frac{1}{98}=\frac{98x}{98}$
Simplify.
$-\frac{1}{98}=x$
Thus, there is no solution.