TITLE: How many solutions are there for the equation $a^x = \log_a x$, where $0 < a < 1$?
QUESTION [5 upvotes]: How many solutions are there for the equation $a^x = \log_a x$, where $0 < a < 1$?

When I first saw this quiz for japanese high school students, I wondered there was only 1 solution for the equation for any $0 < a < 1$.  
But I was wrong:

Then, for what values of $a$ such that $0 < a < 1$ are there 3 solutions for the equation?

REPLY [4 votes]: Too complex for a quiz.
Consider that you look for the zero's of function
$$f(x)=a^x-\frac{\log (x)}{\log (a)}$$ Its derivative is given by
$$f'(x)=a^x \log (a)-\frac{1}{x \log (a)}$$ this cancels at two points given by 
$$x_1=\frac{W_0\left(\frac{1}{\log (a)}\right)}{\log (a)}\qquad \text{and}\qquad x_2=\frac{W_{-1}\left(\frac{1}{\log (a)}\right)}{\log (a)}$$ where appears Lambert function. In the real domain, we need $a \lt e^{-e}$. When this is the case, $f(x_1)>0$ and $f(x_2) < 0$ and in this range $\lim_{x\to 0} \, f(x)=\infty$. So, for $0 < a < e^{-e}$, there are three roots (the first one between $0$ and $x_1$; the second one between $x_1$ and $x_2$; the third one above $x_2$); for $a=e^{-e}$, there is a triple root and for $a>e^{-e}$, there is a single root.
Edit
Since this is an interesting numerical problem, I give you below the three roots  for a faw values of $a$
$$\left(
\begin{array}{cccc}
 a & \text{first root} & \text{second root} & \text{third root} \\
 0.00500 & 0.005883 &  0.256675 &    0.969312 \\
 0.01000 & 0.013093  &  0.277987 &    0.941488 \\
 0.01500 & 0.021585  &  0.292615 &    0.913335 \\
 0.02000 & 0.031462  &  0.304205 &    0.884194 \\
 0.02500 & 0.042894  &  0.314008 &    0.853652 \\
 0.03000 & 0.056133   &  0.322619 &    0.821327 \\
 0.03500 & 0.071532  &  0.330371 &    0.786783 \\
 0.04000 & 0.089601  &  0.337471 &    0.749451 \\
 0.04500 & 0.111117   &  0.344056 &    0.708514 \\
 0.05000 & 0.137359   &  0.350225 &    0.662661 \\
 0.05500 & 0.170721   &  0.356048 &    0.609472 \\
 0.06000 & 0.216898   &  0.361580 &    0.543230 \\
 0.06500 & 0.303124   &  0.366862 &    0.436682 \\
 0.06510 & 0.306379   &  0.366965 &    0.433018 \\
 0.06520 & 0.309837   &  0.367069 &    0.429151 \\
 0.06530 & 0.313538   &  0.367172 &    0.425041 \\
 0.06540 & 0.317536   &  0.367275 &    0.420633 \\
 0.06550 & 0.321911   &  0.367378 &    0.415848 \\
 0.06560 & 0.326787   &  0.367481 &    0.410562 \\
 0.06570 & 0.332376   &  0.367584 &    0.404564 \\
 0.06580 & 0.339098   &  0.367686 &    0.397432 \\
 0.06590 & 0.348099   &  0.367789 &    0.388021 \\
 0.06591 & 0.349246   &  0.367799 &    0.386833 \\
 0.06592 & 0.350471   &  0.367810 &    0.385567 \\
 0.06593 & 0.351791   &  0.367820 &    0.384206 \\
 0.06594 & 0.353233   &  0.367830 &    0.382723 \\
 0.06595 & 0.354836   &  0.367840 &    0.381079 \\
 0.06596 & 0.356672   &  0.367851 &    0.379202 \\
 0.06597 & 0.358881   &  0.367861 &    0.376952 \\
 0.06598 & 0.361865   &  0.367871 &    0.373927
\end{array}
\right)$$
For $a=e^{-e}$, the triple root is $0.367882$.
For the case of a single root
$$\left(
\begin{array}{cc}
 a & \text{ root} \\
 0.10 & 0.399013 \\
 0.15 & 0.436709 \\
 0.20 & 0.469622 \\
 0.25 & 0.500000 \\
 0.30 & 0.528956 \\
 0.35 & 0.557154 \\
 0.40 & 0.585043 \\
 0.45 & 0.612961 \\
 0.50 & 0.641186 \\
 0.55 & 0.669965 \\
 0.60 & 0.699535 \\
 0.65 & 0.730133 \\
 0.70 & 0.762013 \\
 0.75 & 0.795457 \\
 0.80 & 0.830785 \\
 0.85 & 0.868378 \\
 0.90 & 0.908699 \\
 0.95 & 0.952326
\end{array}
\right)$$