Exponent to which the number 10 must be raised to obtain a given number.

- Consider the expression \(\log_{10}(649) ≈ 2.812\).

The characteristic (the integer part of the logarithm) is 2, and the mantissa is 0.812, or 2.812 – 2 = 0.812. - Consider the expression \(\log_{10}(3.779) ≈ 0.577\).

The characteristic (the integer part of the logarithm) is 0, and the mantissa is 0.577, or 0.577 – 0 = 0.577.

### Notation

If \(a\) = 10\(^{n}\), then \(n\) is the common logarithm of \(a\) to the base 10.

This relationship is written as: \(n = \log_{10}(a)\) which is read as “\(n\) is equal to the logarithm of \(a\) to the base 10″.

### Examples

- If log\(_{10}(100) = 2\), then 100 = 10\(^{2}\).
- If log\(_{10}(1000) = 3\), then 1000 = 10\(^{3}\).