Choosing $a$ such that equation $f(x, a)=g(x)$ has only one solution with respect to $x$ Let $f(x, a)$ and $g(x)$ are given functions. We want to find all values of $a$ such that equation $f(x, a)-g(x)=0$ has only one solution with respect to $x$.
For example:

$a \cdot \log_{a}x=x$

or

$ax^4 - e^x=0$


Neat question! For the second, we have the first condition, that $f=0$ for some pair $(x,a)$
or
$$
a=\frac{e^x}{x^4}
$$
Noting that if $x=0$ we don't have any solutions at all.
Then we use a tangency condition of the two curves to guarantee uniqueness, getting a system of two equations in two unknowns, i.e.
$$
f'(x)=0\Rightarrow a=\frac{e^x}{4x^3} 
$$
Combining:
$$
x=4,\\
a=\frac{e^4}{4^4}
$$
For the first equation, use that $$\log_a x =\frac{\ln x}{\ln a}$$
and the same logic as above to get the system
$$
a\frac{\ln x}{\ln a}=x\\
\frac{a}{\ln a}=x\\
\Rightarrow \ln x=1\Rightarrow x=e\Rightarrow a=e
$$
with a's value by inspection.
