# Exponential touch Logarithm

When I write graph for fun I found that there is some value a that make $$y=a^x$$ and $$y=\log_a (x)$$ don't touch each other and there is some value a that make $$y=a^x$$ and $$y=\log_a (x)$$ intersect each other too (2 intersect point). Because of this there is a value a that make they touch each other.

https://sv1.picz.in.th/images/2019/05/03/wKD9ry.jpg

So, I try to find that value by writing graph. Then, I get a = 1.4444... or 13/9. But I have no idea why a = 13/9.

• It is not $\frac {13}9$ but $e^{\tfrac 1e}\approx 1.44466786100977$ instead. – Mohammad Zuhair Khan May 2 at 17:17
• If $a^x=\log_a x \implies x=a^{a^x}$. Solutions of $x = a^x$ are also solutions to $x = a^{a^x}$ so you can achieve some solutions with the Lambert-W function. – Mohammad Zuhair Khan May 2 at 20:25