Given these two examples from my math course:
Example A: $$\log(50) + \log(x/2) = 2 \implies \log 25x = 2 \implies 25x = 10^2 \implies x = 4.$$
Example B: $$\log(72) - \log(2x/3) = 0 \!\implies\! \log(72) = \log(2x/3) \!\implies\! 72 = 2x/3 \!\implies\! 216 = 2x \!\implies \!108 = x$$
My question: Why in Example B, we move $-\log(2x/3)$ over to the other side instead of using the Quotient Property? On the other hand, in Example A we didn't move $\log(x/2)$ to the other side but used the product property instead.
Attempt:
In Example A, if we move $\log(x/2)$ to the other side, we have $$\log(50) + \log(x/2) = 2 \implies \log(50) = -\log(x/2) +2.$$
Now we could raise everything to a common base of $10$ to cancel out the logarithms. So we are left with the following $$50 = -(x/2) + 100.$$
Now we subtract 100 from each side to get
$$-50 = -(x/2) \implies -50 = -(x/2) \implies 100 = 4$$
But as shown in the example $x = 4$.
Why this didn't work?