Math question.

Vitaly1986

Hey people.
Just a quick question.
When we have graph Y = Ln(x) and Y = mX where m is a constant, how to find coordinates of the point where they touch. Basicly solve equation mX = Ln(x).
I think the answer is e^(-1) but can't work out solution atm.
Thanks

Reply 1

Esquire

there exists m for which there is no solution to this equation...

Reply 2

Vitaly1986

and there exist m for which there are 2 solutions but I need coordinates of the point where y = mX and y = Ln(x) touch (Intercept only once).

Reply 3

Speleo

If mx touches lnx once, the gradients of the two functions will be equal.
Grad of lnx at point (p,q) is 1/p, and of mx is m.
So 1/p = m, p = 1/m so q = mx = m(1/m) = 1
So the point at which lnx touches mx only once is (1/m, 1)
Since q = 1, lnx = 1 so x = e
So 1/m = e
So m = 1/e
So the line is y = x/e and the point at which they touch is (e , 1)