Common Logarithm and Natural Logarithm

Common logarithm

A logarithm to base $10$ is called a common logarithm, usually written without showing the base at all:

Common logarithms are the standard choice in most numerical/commercial applications because our number system is base-10, so powers of 10 (and hence base-10 logarithms of round numbers) are especially clean — e.g. $\log 1000=3$ since $10^3=1000$, and $\log 0.01=-2$ since $10^{-2}=0.01$.

Natural logarithm

A logarithm to the special base $e$ (Euler's number, $e\approx2.71828$) is called the natural logarithm, written $\ln N$ or $\log_e N$:

The base $e$ arises naturally in continuous-growth contexts (compound interest compounded continuously, certain statistical and growth models) — for this chapter's scope, the key fact needed is simply that $\ln e = 1$ and $\ln 1 = 0$, exactly like any other base under the two standard values from Section 1.

Quick evaluation using $\ln e^x = x$ …