Finding the number of the real roots of $a^x=g(x)$ where $g(x)$ is the inverse function of $f(x)=a^x$

Asked Oct 17 '13 at 07:32

Active Oct 17 '13 at 07:32

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Question : Let $a$ be a constant which satisfies $0\lt a\lt 1$. Letting $g(x)$ be the inverse function of $f(x)=a^x$, then find the number $N$ of the real roots of $f(x)=g(x)$.

Motivation : This is the question which I created. First, I was thinking about the $a\not=1, a\gt0$ case. Then, I found that the $a\gt 1$ case was easy to solve.

The answer for the $a\gt 1$ case: $N=2$ when $1\lt a\lt e^{\frac 1e}$. $N=1$ when $a=e^{\frac 1e}.$ $N=0$ when $a\gt e^{\frac 1e}$.

Then, I found that the $0\lt a\lt 1$ case was not very easy to solove. Tedious calculations led me to an answer, but I'm not sure if it is correct. (I guess that we need divide $0\lt a\lt 1$ into two parts.) Can anyone help?

asked Oct 17 '13 at 07:32

mathlove

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Have you tried using $a^x=e^{alnx}$ ? – BFD Oct 17 '13 at 07:40
$g(x)=\ln(x)/\ln(a)$, $f(x)=e^{a\ln(x)}$. Letting $z=\ln(x)$, we have to solve $\dfrac{z}{\ln(a)}=e^{az}$. Now the left hand side is a straight line which is decreasing for $0<a<1$ whereas the right hand side is strictly increasing. – Michael Hoppe Oct 17 '13 at 09:08
@BFD: Wait! I think you are wrong. $a^x$is not $e^{a\ln x}$. $a^x=e^{x\ln a}$ is true. – mathlove Oct 17 '13 at 10:13
@MichaelHoppe: I think you also think $a^x$ as $e^{a\ln x}$. $a^x=e^{x\ln a}$ is true. – mathlove Oct 17 '13 at 10:14
@mathlove shame on me ... Of course you're right. – Michael Hoppe Oct 17 '13 at 11:05
Is seems one has to consider $0<a<1/e$ and $1/e<a<1$. May it be that there are up to three intersections? – Michael Hoppe Oct 17 '13 at 11:52
@MichaelHoppe: I'm interested in why you got $\frac 1e$. – mathlove Oct 17 '13 at 14:32
@mathlove The interesting value is $a'=e^{-e}$ -- I had to much logarithms in my trials. For $a<a'$ there seem to be three intersections, for $a\geq a'$ there is only one. How I got this? I've plotted $\ln(x)$ and $\ln(a)e^{x\ln(a)}$ for different values of $a$. – Michael Hoppe Oct 17 '13 at 15:09
@MichaelHoppe: Oh, yes. It is exactly what I guessed. – mathlove Oct 17 '13 at 15:24
let us continue this discussion in chat – Michael Hoppe Oct 17 '13 at 15:40

Finding the number of the real roots of $a^x=g(x)$ where $g(x)$ is the inverse function of $f(x)=a^x$

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