Classification of $A^x=B$ vs $xA^x=B$

Question

Given these two types of a single-variable equation (with $x$ denoting the variable):

• $A^x=B$, which can be solved via logarithm
• $xA^x=B$, which can be solved using Lambert W function

Is there a standard terminology which classifies these types under two different categories?

I'm basically asking - what is the standard class name for each one of these equation types?

Answer 1

Both equations are transcendental equations.

Let $H$ a function of one variable, $x$ a variable and $c$ a constant. For naming the type of an equation $H(x)=c$, the name of the type of the function $H$ is used.

$A^x=B$ is an exponential equation.
An exponential equation is a kind of exponential polynomial equations.

$xA^x=B$ is an exponential polynomial equation.
Only some simple exponential polynomial equations can be solved in closed form.