Can I transform tetration into power by logarithm?

Question

So I was thinking about random maths when I couldn’t sleep last night, and I had an idea.

You can simplify $\ln{(a^b)}$ into $b\ln{(a)}$, and $\ln{(ab)}$ into $\ln{(a)}+\ln{(b)}$.

Observing this, it seems that $\ln$ turns power into multiplication, and multiplication can into addition, and by the sequence, $\ln{(^ba)}$ can be simplified into $\ln^b{(a)}$?

Is my statement true, and why / why not?

Answer 1

$\log_e\left(e^{e^e}\right)= e^e\log_e(e)=e^{e} \not = 1 =\left(\log_e(e)\right)^3$

and more generally $\log{(^ba)} = {^{b-1}a}\,\log(a)$ is usually not $\left(\log(a)\right)^b=\log^b(a)$

Answer 2

No, for example: $$ \mathrm{ln}(^3e) = e^e\mathrm{ln}(e) = e^e \neq \mathrm{ln}^3(e) = 1. $$

Answer 3

No.

Counterexample: $\ln(^32) = \ln(2^4) = 4\ln 2 \neq (\ln 2)^3$.

More generally, the good formula is $\ln (^ba) = \ln (a^{(^{b-1}a)}) = (^{b-1}\!a) \ln a$.