Common Logarithm
The common logarithm, written log(x) or log_10(x), is the logarithm with base 10 and represents the power of 10 needed to equal x.
Formula
\log(x) = \log_{10}(x)
Definition
The common logarithm is a logarithm with base $10$: when "log" is written without a base, it usually means base $10$, asking "what power of $10$ gives this number?" Formally, $\log_{10}(x) = \log(x)$ is the inverse of $10^x$, satisfying $\log(10^x) = x$ and $10^{\log(x)} = x$ for $x > 0$; it relates to the natural logarithm by $\log_{10}(x) = \ln(x)/\ln(10) = \ln(x)/2.302585\ldots$, a constant scaling factor. Historically it was the most computationally convenient logarithm before electronic calculators, though in number theory ln is the natural choice (Mertens' theorem, the prime number theorem), while $\log_{10}$ still appears in estimates of the number of digits of $n!$ and other combinatorial quantities.
Example
$\log(100) = 2$ because $10^2 = 100$, $\log(1000) = 3$, $\log(10) = 1$, $\log(1) = 0$, and $\log(0.01) = -2$ because $10^{-2} = 0.01$. On the Richter scale, $M = \log(I/I_0)$, so a magnitude $6$ earthquake is $10$ times stronger than magnitude $5$; on the pH scale, $\text{pH} = -\log([H^+])$, so pH $4$ is $10$ times more acidic than pH $5$. The number of decimal digits of $n!$ is $\lfloor \log_{10}(n!) \rfloor + 1$, and by Stirling's approximation, $\log_{10}(n!) \approx n\log_{10}(n) - n\log_{10}(e) + 0.5\log_{10}(2\pi n)$.
Key Insight
Base $10$ is convenient because our number system is base $10$: $\log(x)$ roughly tells you how many digits $x$ has, more precisely, the number of digits of $n$ is $\lfloor \log(n) \rfloor + 1$, so common logarithms are especially useful whenever working with powers of $10$ or "orders of magnitude," as in scientific notation. In information theory $\log_2$ is standard (measuring bits), in thermodynamics $\ln$ is standard, and $\log_{10}$ was historically the most computationally convenient; all three are related by constant scaling factors, reflecting that there is really only one logarithm function.