Logarithmic equations and their related exponential equations

Question

I am learning about logarithms and I'd like some examples of the following:

• Examples where the logarithmic value is a different positive integer
• Examples where the logarithmic value is a different negative integer
• Examples where the logarithmic value is a different non-integer fractional value

Answer 1

I suspect the question might not be about natural logarithms. If you allow arbitrary bases, it's easy to construct examples: $\log_b a = c$ if $a = b^c$. So:

$\log_2 8 = 3$

$\log_3 (1/9) = -2$

$\log_4 32 = 5/2$

Answer 2

You do know that the exponential is the inverse bijection of the logarithm, right ? So if you want to find the (unique) value $x >0$ such that $\ln(x)=y$ for some fixed $y \in R$ the answer is $x = e^y$.

It is not exactly your question, but you won't find any value where both $x$ and $y$ are, say, both rational, except if $y=0$ and $x=1$.