Question

Given the exponential function f(x)=a^x
and the inverse of the function, f(x)=log_a(x) where a > 0.
For what values of a do the graphs of f(x)=a^x and inverse f(x), log_a(x) intersect?

Answer 1

looks hard

Answer 2

Using trial and error i found that when a is between 1.1 and 1.44

Answer 3

both intersect at 2 points

Answer 4

im getting 0<a<1.45

Answer 5

look at the attached animation and slider

Answer 6

try this

Answer 7

oh yea thats true

Answer 8

okays so they intersect when 0<a<1.45.

Answer 9

thats right, but how do u solve it analytically ?

Answer 10

Thats the tricky part

Answer 11

\[a^x=\log _{a}x\]
\[(a^x)^{(a^x)}\]

Answer 12

woops i messed up.

Answer 13

the exact solution is :
\[\large 0\lt a \le e^{1/e}\]

Answer 14

idk how to prove it though

Answer 15

\[x=a ^{a^x}\]

Answer 16

okay, i guess its just a trial and error question. lets just move on to the next part of the question.

Answer 17

Describe as much as you can about the point of intersections of the graphs in part a). Well from the graphs i noticed that they both intersect and two points, and the (x,y) values are identical.

Answer 18

*(x,y) values are identical at points of intersection.

Answer 19

thats a very good observation !

Answer 20

Thanks, you know how you gave the exact solution, should i write the exact solution or should i just write 0<a<=1.44

Answer 21

i don't think you want to write the exact solution without giving a proof

Answer 22

and we dont have a proof, so forget about e^(1/e)

Answer 23

Okay, so I wrote, from process of trial and error, the graph of f(x) and inverse f(x), intersect when \[0<a \le1.44\] is that fine?

Answer 24

Looks good to me !

Answer 25

Alright thanks :D

Answer 26

np :) let me know if you find an analytic solution

Answer 27

Sure thing :)