Logarithm and exponential inverse (base 2)

Question

if we have a function as follows

$$\log_2(1+ S) \leq T$$

Which of the following is correct

$$S\leq e^T-1$$ or $$S\leq 2^T-1$$

When should each be used, i.e. with what logarithm base? Thanks

Answer 1

If you have $\log_a(1+S)\leq T$, it means that $1+S \leq a^T$. So $S \leq a^T -1.$ Thus, this the general formula for your problem. The first case is $a=e$, so, $S \leq e^T -1.$ The second is $a=2$, so, $S \leq 2^T -1.$