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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?

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    $\begingroup$ I think $\log_{10} x=\dfrac{\log_e x}{\log_e 10}$ is already a nice definition. $\endgroup$ – Eclipse Sun Apr 22 '15 at 15:38
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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: $log_{10}x=(log_ex)/(log_e10)$.

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  • $\begingroup$ I didn't understand the point. Can you explain it further? $\endgroup$ – Groups Apr 22 '15 at 15:42

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