This was inspired by Fun logarithm question, because it made me remember a question I accidentally asked on a quiz some time ago. It was suppose to have both log bases the same, 3 or 5. $$\log_{5}\left(x+3\right)=1-\log_{3}\left(x-1\right)$$
After apologizing to my students, I talked to some people about it and we could not find an analytical solution… other than to realize that $x=2$ is a solution by just trying it. So, my questions are: Is there an analytical solution to this specific problem? And, more importantly, why do variables in exponents/logarithms that are seemingly easy to state produce such difficult problems? I would like some insight into the second question more than the first, as answering the second will also answer the first, I think.
[Math] Why do logarithms produce such difficult problems
logarithmssoft-question
Best Answer
If you put everything in a common base, let's say 5, this equation is equivalent to $$(x+3)(x-1)^c=5 $$, with c = $\log_53 \approx 0.68261$. Solving expressions with polynomials is usually easy. I would guess that the problem here is that the exponents are not integer, but real, which makes everything harder.