The logarithmic function with base $e$ is the (set theoretic) inverse of exponential function $e\colon \mathbb{R}\rightarrow (0,\infty)$. This can also be defined using integration as: $\log\colon (0,\infty)\rightarrow \mathbb{R}$, $\log x= \int_1^x \frac{1}{t} dt$.
Question: Can we define logarithm at base $10$ using integration?
Best Answer
The logarithm of base then can be integrated but the base has to be changed to $e$. When you have a function, lets say $f(x)=logx$ then the base can be changed from 10 to $e$. This is done using the rule $log_mx=(log_nx)/(log_nm)$. So changing from base 10 to base e would look as follows: $log_{10}x=(log_ex)/(log_e10)$.