CONTEXT:
I'm a Calculus I student and recently realized (While trying to solve an unrelated problem) that while I can add, substract, multiply or divide pretty much any two real numbers by hand to a pretty reasonable accuracy (< 0.05% error if I use three digits after the decimal).
I literally wouldn't know how to solve something like $(5.42)^{3.58}$ or $π^{π-1}$ to even 1 decimal digit (after the comma) of accuracy. I did learn about Taylor polynomials as approximations of functions recently, and have found them really useful for approximating (by hand), something like $\sin(0.64)$ or $\cos(0.13)$, but I haven't found a way to use them for exponential equations ($a^x$) without knowing the natural log of the number beforehand.
I searched online for how to solve $a^b$, but only found solutions for "clean numbers", or fractional exponents that end up simplifying the problem (via roots): e.g. $32^{0.4}$ = $32^\frac 25 = (32^{\frac 15})^2 = 2^2 = 4.$
Question: Is there a method to solving by hand something like $a^b$, where a and b are both real numbers, to a reasonable (Let's say < 0.1%) accuracy ?