Simplifying $\log$ expression

Question

I need to simplify the following expression:

$$1+\frac{\log_2\left(\frac{4}{5}\left(x+3\right)\right)}{\log_2\left(\frac{5}{4}\right)}\to \frac{\log_2\left(x+3\right)}{\log_2\left(\frac{5}{4}\right)}$$

It's been a while since I did logs, but I cannot figure out the trick here.

I think I might be able to use the fact that $\log_2(4/5)=-\log_2(5/4)$, but do not see any immediate results.

Answer 1

You have to use $\log(ab)=\log(a)+\log(b)$.

Answer 2

Use $$\log_2\frac{4}{5}+\log_2\frac{5}{4}=0.$$

Answer 3

Here's a hint:

$$1+\frac{\log_2\left(\frac{4}{5}\left(x+3\right)\right)}{\log_2\left(\frac{5}{4}\right)} = 1+\frac{\log_2{\frac45}+\log_2(x+3)}{-\log_2\frac{4}{5}}=1-1-\frac{\log_2(x+3)}{\log_2\frac45}=\dots? $$

Answer 4

$$ 1+\frac {\log_2 (\frac{4}{5} (x+3))} {\log_2 (\frac{5}{4})}= \frac {\log_2 \frac{5}{4} + \log_2 \frac{4}{5} + \log_2 (x+3)} {\log_2\left(\frac{5}{4}\right)}= \frac{\log_2 1 + \log_2\left(x+3\right)}{\log_2\left(\frac{5}{4}\right)} = \frac {\log_2(x+3)} {\log_2(\frac 5 4)} $$

using $\log(ab)=\log(a)+\log(b)$, as suggested in another answer.