$f(x) = \ln(x+1)$ transformation, transform it to $f(x) = g(x)\ln(x)$

Question

$f(x) = \ln(x+1)$

Is there a way to transform the Equation above to a simpler one, that will include only $f(x) = g(x)\ln(x)$ kind of function?

Answer 1

If: $$\ln (x+1) = f(x) \ln x$$

then:

$$f(x) = \frac{\ln(x+1)}{\ln x}$$

This cannot be simplified. You could write $f(x) = \log_x(x+1)$, but we generally use logarithms with constant bases (hopefully $e$, that's the simplest base) so it is not a good idea to do so.