The question asks me to find the minimum value of $a$, where $a > 1$, that $f(x) = a^x$ and $g(x) = \log_a x$ have a common tangent.
I solved for the derivatives of the two functions, and got $$f'(x) = \ln (x) * a^x $$ $$ g'(x) = \frac{1}{x*\ln(a)}$$ $$\ln (x) * a^x = \frac{1}{x*\ln(a)}$$
I also know that at the tangent, the two functions must equal each other. So $$a^x = \log_a x$$
But from there, I unfortunately don't know how to continue. What steps should I take next in finding the $a$ where the two functions have a common tangent? I did try substituting $\log_a x$ in for $a^x$ but that didn't lead me anywhere.
You can consider the equation $a^x=x$ taking the logarithm on both sides we get
$\ln(a)=\frac{\ln(x)}{x}$ now define $f(x)=\frac{\ln(x)}{x}$ and use calculus. For $x=e$ we get the maximum values, $f(e)=\frac{1}{e}$ so we get $$\ln(a)=\frac{1}{e}$$ or $$a=e^{1/e}$$