I created an equation a bit ago where I knew the answer, but not how to solve it.
Equation: $$x = 10^{x-1}$$
Answer: $x = 1$
I can not see to find any documentation related to this problem. I know that the equation can be written as $x = a^{x-1}$ and the answer will still be $x = 1$. The only reason I know the answer is because I can plug it in and I made it around the answer. I would love to know the steps behind how to solve this equation.
You can rewrite your equation as
$$-x\ln 10 e^{-x\ln 10}=-\frac{\ln 10 }{10},$$ and then use Lambert W function to solve the obtained equation: $-x\ln 10 = W\left(-\frac{\ln 10 }{10}\right)$. As you can read on wiki, for some values of $z$ $W(z)$ is multiply defined, hence one one branch you obtain $\forall z>0 W(z\ln z) = \ln z$, which in our case gives $x=1$. Other branches of $W$ will give other roots of this equation.