Logarithms are a convenient way to express large numbers. (The base-10 logarithm of a number is roughly the number of digits in that number, for example.) Slide rules work because adding and subtracting logarithms is equivalent to multiplication and division. (This benefit is slightly less important today.)

In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g., since 1000 = 10 × 10 × 10 = 103, the “logarithm base 10” of 1000 is 3, or log10(1000) = 3. The logarithm of x to base b is denoted as log b(x), or without parentheses, log b x, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation.

More generally, exponentiation allows any positive real number as base to be raised to any real power, always producing a positive result, so log b(x) for any two positive real numbers b and x, where b is not equal to 1, is always a unique real number y.

## Abbas Khan

Logarithms are a convenient way to express large numbers. (The base-10 logarithm of a number is roughly the number of digits in that number, for example.) Slide rules work because adding and subtracting logarithms is equivalent to multiplication and division. (This benefit is slightly less important today.)

In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g., since 1000 = 10 × 10 × 10 = 103, the “logarithm base 10” of 1000 is 3, or log10(1000) = 3. The logarithm of x to base b is denoted as log b(x), or without parentheses, log b x, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation.

More generally, exponentiation allows any positive real number as base to be raised to any real power, always producing a positive result, so log b(x) for any two positive real numbers b and x, where b is not equal to 1, is always a unique real number y.