thanks for clicking on my topic. I have only just finished high school so apologies if this is a stupid question in advance!
I was messing around with the equation $$x^a - b^x = 0$$
I didn't find this from anywhere, was just thinking about it as a further generalisation of $$x^n - n^x = 0$$
So I did the following: $$x^a - b^x = 0$$
$$x^a = b^x$$
$$\ln(x^a) = \ln(b^x)$$
$$a \cdot\ln(x) = x\cdot \ln(b)$$
$$x/\ln(x) = a/\ln (b)$$
The left-hand side above is an asymptotic approximation for the number of primes up to $x$, by the prime number theorem. So by substituting $a$ and $b$ into the right-hand side for a given case, a range for $x$ could be determined based on how many primes precede it. This is obviously useless because computers can solve this already, but I thought maybe it shows some possible link between the prime numbers and exponential and polynomial function intersections. It probably means nothing but I wanted to post this anyway, just to see what others think. Thanks!