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## Definitions

### General

The decimal (also called decadic) logarithm of a number x, is a power p so that 10^{p} = x. In modern notation decimal logarithm is written as \(\log_{10}{x}\) or lg(x). Any positive real number has a decimal logarithm. The decimal logarithm of 1 is 0, while the decimal logarithm of 0 approaches negative infinity.

The graph of the decimal logarithm function is shown in the figure below. It is a monotonic function.

### Properties

The derivative of the decimal logarithm function is:

\[ \left(\log_{10} x\right)' = \frac{1}{x}\frac{1}{\ln{10}} \approx \frac{0.4342945}{x} \]

The integral of the decimal logarithm function is given by:

\[ \int \log_{10} x\, \mathrm{d}x = \frac{x\ln{x}-x}{\ln{10}} +C \]