Logarithms and exponentials in LaTeX
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The natural logarithm is \ln x, the common logarithm is \log x or \log_{10} x, and any other base goes in a subscript: \log_b x. A long exponent of e is easier to read when written with \exp and parentheses.
Logarithms
\ln e = 1, \quad \log_{2} 8 = 3\ln and \log are operator names: set upright, with a thin space before the argument. The base is an ordinary subscript, so a base with more than one character needs braces, as in \log_{10}.
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Change of base
\log_b x = \frac{\ln x}{\ln b}The ratio of two natural logarithms is how you compute any base on a calculator, and \frac keeps both logarithms at a readable size.
Logarithm rules
\log_b (xy) = \log_b x + \log_b yPut the argument in brackets as soon as it has more than one factor. LaTeX would typeset \log_b xy without complaint, but a reader cannot tell where the argument ends.
Exponential function
e^{x}, \quad e^{-\lambda t}, \quad \mathrm{e}^{x}Braces around the exponent are needed once it has more than one character. Some style guides set the constant e upright as \mathrm{e}; choose one style for the whole document.
Radioactive decay
N(t) = N_0 \exp\left(-\frac{t \ln 2}{T_{1/2}}\right)A fraction in the exponent of e would shrink to a tiny size. \exp keeps the argument on the main line, and \left( \right) lets the brackets grow with the fraction.
Derivative of ln
\frac{d}{dx} \ln x = \frac{1}{x}A line that appears in every set of calculus notes. \frac{d}{dx} works as an operator written directly in front of \ln.
Common mistakes
- log without a backslash: the letters are set in italics.
- \log2 8 instead of \log_2 8: without the underscore the 2 is not a base, just a factor.
- lg or lb typed as plain letters: they have no built-in command. Define them with \operatorname{lg} x.
See also
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