I was playing around in Desmos, and amidst that I became interested in how the graphs of exponentials and quadratics interact. For $0<a<1$ everything was as expected and nothing intriguing happened. However, for $a>1$ I noticed how $x^2=a^x$ had two real solutions for $1<a<2$. After playing around with the slider for a bit more I saw that this was true till about $a=2.1$ For any value of $a$ after that the graphs never met for $x>0$. So, I thought of finding the exact value of $a$ for which the two graphs just touch, or in other words, have exactly one solution.
At first I thought it would be trivial to find out at exactly what value of $a$ this happened. But when I actually sat down to try the problem, I realised I had no idea what to do. At first I thought the derivatives of the two functions would be equal when they touch, and even though that's true I found that the derivatives could be equal at other places too (not to say I have no idea how to solve $2x=a^x\ln a$ either)
I'm sure I am again making a big fuss out of something trivial. But, I just can't seem to see how we can solve this (with elementary maths). Can someone please point me in the right direction? Thanks