Is it possible to express $\ln(x+1)$ in terms of $\ln(x)$?

Question

Is there a way to express $\ln (x+1)$ as $A\ln x + B$, where $A$ and $B$ are some expressions in $x$ that do not themselves include logarithmic terms, $A\neq 0$, and $x>1$?

Answer 1

$$\ln(x+1)=\ln(x)-\sum_{k=1}^\infty\frac{(-1)^kx^{-k}}{k}, \text{for $|x|>1$}$$