Work in Progress.

When $k = 1$ this equation has the solution $x = \log_b( \frac C A)$.

When $k \ne 1$ the equation can be solved so that $x = \frac 1k \log_b( \frac C A)$.

Comments: You can move the (red) point labelled x on the left axis of the mapping diagram to a position where the arrow head points to $f(x) = C$.

Check the box and the diagram will show the solution.

You can use the sliders to investigate other examples by changing the base, $A,h$, and $k$, as well as the point $C$ on the target axis.