Answer
False.
Work Step by Step
We are given the following expression:
$\log(x-2)\stackrel{?}{=}\frac{\log x}{\log 2}$
We might be tempted to apply the quotient property of logarithms to the left or right side of our expression. However, recall that the quotient property has a different form from our expression:
$\log_b{\frac{m}{n}}=\log_b{m}-\log_b{n}$
Notice that the minus sign is between the values $m$ and $n$ on the left side and the division occurs between the logs on the right side. However, we need the minus sign to be between the logs and the division sign to be between the values. Thus we can not apply the quotient rule to simplify our expression.
Thus we suspect that the expression is false. We can confirm this by plugging in any number, e.g. $12$:
$\log(12-2)\stackrel{?}{=}\frac{\log 12}{\log 2}$
$\log(10)\stackrel{?}{=}\frac{\log 12}{\log 2}$
$1\ne3.58$
Thus we see that the expression is False.
