Exp = log

Calculus Level 5

Let $f(b)$ denote the number of real solutions $x$ satisfying $b^x = \log_b x.$

Find the closed form of the integral $\displaystyle \int_0^\infty f(b) \, db$ to 3 decimal places.

1 solution

Jeremy Galvagni
Apr 22, 2018

There are $3$ solutions on $(0,e^{-e})$

There is $1$ solution on $[e^{-e},1]$

There are $2$ solutions on $(1,e^{1/e})$

There is $1$ solution at $e^{1/e}$

There are $0$ solutions on $(e^{1/e},\infty)$

The integral then is just the areas of a few rectangles:

$3*e^{-e} + 1*(1-e^{-e}) + 2*(e^{1/e}-1) = 2e^{-e}+2e^{1/e}-1 = 2.021311794$

Why are there 3 solutions in 0 to e^-e?

Aditya Gupta - 2 years, 8 months ago

https://www.desmos.com/calculator/frrhlzpgkb

Jeremy Galvagni - 2 years, 8 months ago