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In mathematics, the logarithm of a number to a given base is the power or exponent to which the base must be raised in order to produce the number.

For example, the logarithm of 1000 to the base 10 is 3, because 10 raised to the power of 3 is 1000; the base 2 logarithm of 32 is 5 because 2 to the power 5 is 32.

The logarithm of x to the base b is written log_b(x) or, if the base is implicit, as log(x). So, for a number x, a base b and an exponent y,

${\text{ if }}b^{y}=x,{\text{ then }}y=\log _{b}(x)\,.$

An important feature of logarithms is that they reduce multiplication to addition, by the formula:

$\log(xy)=\log x+\log y\,.$

That is, the logarithm of the product of two numbers is the sum of the logarithms of those numbers. The use of logarithms to facilitate complex calculations was a significant motivation in their original development.

Properties of the logarithm

Template:Main article When x and b are restricted to positive real numbers, log_b(x) is a unique real number. The magnitude of the base b must be neither 0 nor 1; the base used is typically 10, e, or 2. Logarithms are defined for real numbers and for complex numbers.

The major property of logarithms is that they map multiplication to addition. This ability stems from the following identity:

$b^{x}\times b^{y}=b^{x+y}\ ,$

which by taking logarithms becomes

$\log _{b}\left(b^{x}\times b^{y}\right)=\log _{b}\left(b^{x+y}\right)$ $\ =x+y=\log _{b}\left(b^{x}\right)+\log _{b}\left(b^{y}\right).\$

For example,

$4=2^{2}\,\Rightarrow \,\log _{2}(4)=2\,,$

$8=2^{3}\,\Rightarrow \,\log _{2}(8)=3\,,$

$\log _{2}(32)=\log _{2}(4\times 8)=\log _{2}(4)+\log _{2}(8)=2+3=5\,.$

A related property is reduction of exponentiation to multiplication. Using the identity:

$c=b^{\log _{b}(c)}\ ,$

it follows that c to the power p (exponentiation) is:

$c^{p}=\left(b^{\log _{b}(c)}\right)^{p}=b^{p\log _{b}(c)}\ ,$

or, taking logarithms:

$\log _{b}\left(c^{p}\right)=p\log _{b}(c)\ .$

In words, to raise a number to a power p, find the logarithm of the number and multiply it by p. The exponentiated value is then the inverse logarithm of this product; that is, number to power = b^product.

For example,

$\log _{2}(64)=\log _{2}(4^{3})=3\log _{2}(4)=6\,.$

Besides reducing multiplication operations to addition, and exponentiation to multiplication, logarithms reduce division to subtraction, and roots to division. For example,

$\log _{2}(16)=\log _{2}\left({\frac {64}{4}}\right)=\log _{2}(64)-\log _{2}(4)=6-2=4\,,$

$\log _{2}({\sqrt[{3}]{4}})={\frac {1}{3}}\log _{2}(4)={\frac {2}{3}}\,.$

Logarithms make lengthy numerical operations easier to perform. The whole process is made easy by using tables of logarithms, or a slide rule, antiquated now that calculators are available. Although the above practical advantages are not important for numerical work today, they are used in graphical analysis (see Bode plot).