Identities of logarithms.

Question

In this question I learned that $$\lim_{k\rightarrow\infty}\left(\sum_{i=0}^k\frac{1}{2i+1}-\sum_{i=1}^k\frac{1}{2i}\right)=\ln2$$Which was unexpected to me but the solution is simple. I also found that $$\sum_{k\ge1}\frac{1}{k2^k}=\ln2$$In an answer I wrote which I couldn't find the link of. So I wonder if there are any other identities of not just $\ln2$, but also other logarithms like $\ln 3$ or $\log_22$. So post any identities relating to the logarithms of specific values here! (Please don't post trivial appearances like $\ln3=\ln3$).

Edit: I am asking for identities of logarithms, which definitely are facts, am I right? Please reopen the question.

Answer 1

Arguably one of the most interesting appearances of a logarithm is as an integal of a much simpler function: $\displaystyle\int_1^x \frac1t\,dt = \ln x$, and more generally $\displaystyle\int_1^x \frac ct\,dt = \log_b x$ where $b=e^{1/c}$.

Answer 2

Consider the alternating harmonic series: $$\sum_{n=1}^{+\infty}\frac{(-1)^{n-1}}{n} $$ and recall the Taylor expansion of $\log(1+x)$: $$\log(1+x)=\sum_{n=1}^{+\infty}\frac{(-1)^{n-1}}{n}x^n \quad\quad\text{for all}\quad |x|\le 1 \quad\text{and}\quad x\ne-1$$ Hence you have $$\sum_{n=1}^{+\infty}\frac{(-1)^{n-1}}{n}=\log 2 $$