simplify a log expression

Question

I met a problem, I don't know if this term can be simplified properly?

$$e^{ (\ln ax^{b})^{c}}$$

since the ln term with power of c is hard to cope with, thanks for any help!

Answer 1

It's $e^{\ln(ax^b)*\ln(ax^b)^{c-1}} = (ax^b)^{(\ln(ax^b))^{c-1}}$ It doesn't simplify any more than that unfortunately.

Answer 2

No, there's generally nothing productive to do here, unless you know that $c$ is some particular nice number such that $0$ or $1$ ...

It may be useful to rewrite the expression in parentheses to $\ln a + b\ln x$, but whether that is actually progress depends on what you eventual goal is.

Answer 3

You can't simplify it much. You can rewrite it as $e^{{(\ln ax^b)}^{c-1}(\ln ax^b)}$ and then as $(\ln ax^b)^{{(\ln ax^b)}^{c-1}}$ but that's pretty much all you can do here.