Editing Logarithm (section)

===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>
|}