The discussion revolves around the equation a^x = x, specifically exploring the conditions under which this equation has a single solution for a > 1. Participants analyze various values of a and their implications on the solution.
Participants express differing levels of understanding regarding the derivation of x = 1/ln(a) and the implications of the derivative, indicating that the discussion remains unresolved on these points.
Some assumptions regarding the behavior of the function f(x) and the conditions for a single solution may not be fully explored, leaving room for further clarification.
Vali said:If the equation a^x=x with a>1 has one solution then:
A)a=1/e
B)a=e
C)a=e^(1/e)
D)a=e^e
E)1/(e^e)
The right answer is C.I tried to derivate then to resolve f'(x) but didn't work
MarkFL said:I would write:
$$f(x)=a^x-x=0$$
Hence:
$$f'(x)=a^x\ln(a)-1=0$$
These imply:
$$x=\frac{1}{\ln(a)}=\log_a(e)\implies a^x=e$$
And so:
$$\ln(a)=\frac{1}{e}\implies a=e^{\frac{1}{e}}$$
Vali said:Thank you for your response.I don't understand why x = 1/lna
from a^xlna-1=0 => a^x=1/lna; why x = 1/lna ?