Common Logarithm
Functions & Advanced AlgebraThe 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 you see "log" without a base written, it usually means log base $10$. It asks: "What power of $10$ gives me this number?"
Example
$\log(100) = 2$ because $10^2 = 100$. $\log(1000) = 3$. $\log(10) = 1$. $\log(1) = 0$. $\log(0.01) = -2$ because $10^{-2} = 0.01$.
Key Insight
Base $10$ is convenient because our number system is base $10$. $\log(x)$ roughly tells you how many digits $x$ has: $\log(100) = 2$, and $100$ has $3$ digits. More precisely, the number of digits of $n$ is $\lfloor \log(n) \rfloor + 1$.
Definition
The common logarithm $\log_{10}(x) = \log(x)$ is the inverse of $10^x$. It satisfies $\log(10^x) = x$ and $10^{\log(x)} = x$ for $x > 0$. On scientific calculators, the "LOG" button computes this. Useful for orders-of-magnitude comparisons.
Example
The Richter scale: $M = \log(I/I_0)$. An earthquake of magnitude $6$ is $10$ times stronger than magnitude $5$. The pH scale: $\text{pH} = -\log([H^+])$. pH $4$ is $10$ times more acidic than pH $5$.
Key Insight
Common logarithms are useful whenever you are working with powers of $10$ or "orders of magnitude." Scientific notation ($3.7 \times 10^5$) uses base $10$, so log immediately reads off the exponent.
Definition
The common logarithm $\log_{10}(x) = \ln(x)/\ln(10) = \ln(x)/2.302585\ldots$ It is related to the natural logarithm by a constant factor. In number theory, Mertens' theorem and the prime number theorem are naturally stated with ln, but $\log_{10}$ appears in estimates of the number of digits of $n!$ and other combinatorial quantities.
Example
The number of decimal digits of $n!$ is $\lfloor \log_{10}(n!) \rfloor + 1$. By Stirling: $\log_{10}(n!) \approx n\log_{10}(n) - n\log_{10}(e) + 0.5\log_{10}(2\pi n)$.
Key Insight
In information theory, $\log_2$ is standard (measuring bits). In thermodynamics, ln is standard. $\log_{10}$ is historically the most computationally convenient before electronic calculators. All three are related by constant scaling factors.