Identify elements involved in a log product

Question

Since we know that $\log_2 ab = \log_2 a + \log_2 b$, is there a way to figure out numerical values of $a$ and $b$ (or even $\log_2 a$ and $\log_2 b$) if we are just given the value of $\log_2 ab$?

Answer 1

The right-Hand side can be written as

$$\frac{\ln(a)+\ln(b)}{\ln(2)}$$ and $b$ is given by, $$b=e^{p\ln(2)-\ln(a)}$$ and this is $$b=\frac{2^p}{a}$$ where $$p=\log_{2}{ab}$$

Answer 2

Not uniquely. We have $$ \log_2(ab) = C \quad \Leftrightarrow \quad ab = 2^C, $$ and the latter is solved by infinitely many pairs $(a,b)$.

However, if you also know $a$ then you can find $b$ and vice versa.