Show that for a ≥ e, there exists at least one line that is simultaneously tangent to both f(x) = ax and
g(x) = loga x.
I have gotten it to 2 eqn.s and 3 unknowns.
f(x) = a^x,
f'(x)=a^x ln a
g(x) = log_a x = ln x / ln a
g'(x) = 1 / (x ln a)
The line has slope m and is tangent to
f and g at x1 and x2 respectively with
slope m, so:
f'(x1) = g'(x2)
a^x1 ln a = 1 / (x2 ln a) [eq. 1]
[g(x2) - f(x1)] / ( x2 - x1) = f'(x1)
(ln x2 / ln a - a^x1) / (x2 -x1) = a^x1 ln a [eq. 2]
[g(x2) - f(x1)] / ( x2 - x1) = g'(x2)
(ln x2 / ln a - a^x2) / (x2 -x1) = 1/ (x2 ln a) [eq. 3]
But, there are really only two independent equations here.
The next step, I think, is to solve for the y intercept of the tangent
line which would add two more equations but only one more
unknown.
I think the solution will look a bit like this:
https://www.desmos.com/calculator/vguj6i4di1
Edited on January 18, 2025, 11:58 pm
just a start...
