Answer
False.
A possible true statement is:
$\displaystyle \log(x+3)-\log(2x)=\log\frac{x+3}{2x}$
Work Step by Step
Applying the quotient rule to the LHS,
$\displaystyle \log(x+3)-\log(2x)=\log\frac{x+3}{2x}\neq $ RHS
Thus, the statement is false.