I learned a fast way to get $x$ in $\ln(x)$ using ($n= x-1$ btw)
$$\ln(x)=2\left(\frac{1}{2n+1}+\frac{1}{3(2n+1)^3}+\frac{1}{5(2n+1)^5}+\frac{1}{7(2n+1)^7}+\frac{1}{9(2n+1)^9}...\right) $$
from wiki, which basically converts any base (i.e. $x$), like $2,$ into $e$ (base $2.71828..).$
But given something like $3^n=750$ or $2^n=1000,$ I still don't know how to get "n" by hand, and i don't understand how calculators are getting such high precision answers. $30$ decimals of precision must be coming from somewhere, and I think its math, because its too big to fit on a table, and since my computer doesn't crash or even blink when i get an answer, the number of steps can't possibly be in the millions, or likely even the thousands.
What's going on? How do I get $n,$ once I have $\ln(x)$ solved?
If you want to perform a change of basis, from base $a$ to $e$. Where $a>0$. $$\log_ax = \frac{\ln(x)}{\ln(a)}$$ And using your expansion this should work. Where in your examples $a=2$ or $a=3$