$c= (a^{x}-b^{x})$ where $a$,$b$ and $c$ are known real constants. Solve for $x$.

Question

I tried taking $\log$ on both side but i ended with $\log(a^{x}-b^{x})$ which is difficult to solve. Does anybody has idea how to solve the above equation for $x$.

Answer 1

I could try to simplify it a little:$$c=a^x-b^x=a^x-a^{x\log_a(b)}$$$$a^x=u$$$$c=u-u^{\log_a(b)}$$$$0=u^{\log_a(b)}-u+c$$From here, you might notice that we can't solve for $u$ because we can't solve the following general problem:$$0=x^n+ax+c$$If we could solve the above problem for any $n$, then we'd probably have things like algebraic solutions to quintic polynomials and such.

In fact, we pretty much can only solve this if:$$0\le n\le4$$And if $n$ is an integer.

We see that $\log_b(a)$ will most likely not be an integer and it will most likely not be between $4,0$.