Editing Logarithm (section)

==Logarithmic identities==
{{Main|List of logarithmic identities}}

Several important formulas, sometimes called ''logarithmic identities'' or ''logarithmic laws'', relate logarithms to one another.<ref>All statements in this section can be found in {{Harvard citations|last1=Downing| first1=Douglas |year=2003|loc=p. 275|nb=yes}} or {{Harvard citations|last1=Kate|last2=Bhapkar|year=2009|loc=p. 1-1|nb=yes}}, for example.</ref>

===Product, quotient, power, and root===
The logarithm of a product is the sum of the logarithms of the numbers being multiplied; the logarithm of the ratio of two numbers is the difference of the logarithms. The logarithm of the {{Mvar|p}}-th power of a number is {{Mvar|p}}&nbsp;times the logarithm of the number itself; the logarithm of a {{Mvar|p}}-th root is the logarithm of the number divided by {{Mvar|p}}. The following table lists these identities with examples. Each of the identities can be derived after substitution of the logarithm definitions <math>x = b^{\, \log_b x}</math> or <math>y = b^{\, \log_b y}</math> in the left hand sides. In the following formulas, {{tmath|x}} and {{tmath|y}} are [[positive real numbers]] and {{tmath|p}} is an integer greater than 1.

{| class="wikitable plainrowheaders"
|+ Product, quotient, power, and root identities of logarithms
|-
! scope="col" | Identity
! scope="col" | Formula
! scope="col" | Example
|-
! scope="row" | Product
| <math display="inline">\log_b(x y) = \log_b x + \log_b y</math>
| <math display="inline">\log_3 243 = \log_3 (9 \cdot 27) = \log_3 9 + \log_3 27 = 2 + 3 = 5</math>
|-
! scope="row" | Quotient
| <math display="inline">\log_b \!\frac{x}{y} = \log_b x - \log_b y</math>
| <math display="inline">\log_2 16 = \log_2 \!\frac{64}{4} = \log_2 64 - \log_2 4 = 6 - 2 = 4</math>
|-
! scope="row" | Power
| <math display="inline">\log_b\left(x^p\right) = p \log_b x</math>
| <math display="inline">\log_2 64 = \log_2 \left(2^6\right) = 6 \log_2 2 = 6</math>
|-
! scope="row" | Root
| <math display="inline">\log_b \sqrt[p]{x} = \frac{\log_b x}{p}</math>
| <math display="inline">\log_{10} \sqrt{1000} = \frac{1}{2}\log_{10} 1000 = \frac{3}{2} = 1.5</math>
|}

===Change of base===<!-- This section is linked from [[Mathematica]] -->
The logarithm {{math|log<sub>''b''</sub>&thinsp;''x''}} can be computed from the logarithms of {{mvar|x}} and {{mvar|b}} with respect to an arbitrary base&nbsp;{{Mvar|k}} using the following formula:{{refn|group=nb|''Proof:''
Taking the logarithm to base {{mvar|k}} of the defining identity <math display=inline> x = b^{\log_b x},</math> one gets
<math> \log_k x = \log_k \left(b^{\log_b x}\right) = \log_b x \cdot \log_k b.</math>
The formula follows by solving for <math>\log_b x.</math>}}
<math display="block"> \log_b x = \frac{\log_k x}{\log_k b}.</math>

Typical [[scientific calculators]] calculate the logarithms to bases 10 and {{mvar|[[e (mathematical constant)|e]]}}.<ref>{{Citation | last1=Bernstein | first1=Stephen | last2=Bernstein | first2=Ruth | title=Schaum's outline of theory and problems of elements of statistics. I, Descriptive statistics and probability | publisher=[[McGraw-Hill]] | location=New York | series=Schaum's outline series | isbn=978-0-07-005023-5 | year=1999 | url=https://archive.org/details/schaumsoutlineof00bern }}, p.&nbsp;21</ref> Logarithms with respect to any base&nbsp;{{mvar|b}} can be determined using either of these two logarithms by the previous formula:
<math display="block"> \log_b x = \frac{\log_{10} x}{\log_{10} b} = \frac{\log_{e} x}{\log_{e} b}.</math>

Given a number {{mvar|x}} and its logarithm {{math|1=''y'' = log<sub>''b''</sub>&thinsp;''x''}} to an unknown base&nbsp;{{mvar|b}}, the base is given by:

<math display="block"> b = x^\frac{1}{y},</math>

which can be seen from taking the defining equation <math> x = b^{\,\log_b x} = b^y</math> to the power of <math>\tfrac{1}{y}.</math>