Logarithm

In mathematics, a logarithm is a function that gives the in the equation bn = x. It is usually written as logb x = n. For example:

Uses of logarithms

Logarithms are useful in solving equations in which exponents are unknown. They have simple derivatives, so they are often used as the solution of integrals. Furthermore, various quantities in science are expressed as logarithms of other quantities; see logarithmic scale for an explanation and a list.

Related Topics:
Derivative - Integral - Logarithmic scale

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Exponential functions

Sometimes (especially in the context of analysis) it is necessary to calculate arbitrary exponential functions f(x)^x using only the natural exponent e^x:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

f(x)^x = e^{log(f(x)^x)}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

= e^{xlog(f(x))}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Easier computations

Logarithms switch the focus from normal numbers to exponents. As long as the same base is used, this makes a few operations easier:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Before electronic calculators, this made difficult operations on two numbers much easier. One simply found the logarithms of both numbers (multiply and divide) or the first number (power or root, where one number is already an exponent) in a table of common logarithms, performed a simpler operation on those, and found the result on a table. Slide rules performed the same operations using logarithms, but faster and with lower precision than using tables. Other tools for performing multiplications before the invention of the calculator include Napier's bones and mechanical calculators (see history of computing hardware).

Related Topics:
Electronic calculator - Common logarithm - Slide rule - Napier's bones - History of computing hardware

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

In abstract algebra, this property of the logarithm functions can be summarized by noting that any logarithm function with a fixed base is a group isomorphism from the group of strictly positive real numbers under multiplication to the group of all real numbers under addition.

Related Topics:
Abstract algebra - Group isomorphism - Group - Real number

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Calculus

To calculate the derivative of a logarithmic function, the following formula is used

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: rac{d}{dx} log_b(x) = rac{1}{x ln(b)} = rac{log_b(e)}{x}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

where ln is the natural logarithm, i.e. with base e. Letting b = e:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: rac{d}{dx} ln(x) = rac{1}{x}, qquad int rac{1}{x} ,dx = ln(x) + C

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

One can then see that the following formula gives the integral of a logarithm

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: int log_b(x) ,dx = x log_b(x) - rac{x}{ln(b)} + C = x log_b left(rac{x}{e} ight) + C

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

See also: , list of integrals of logarithmic functions

~ ~ ~ ~ ~ ~ ~ ~ ~ ~