maths > exponents

Logarithm : An inverse of Exponent

what you'll learn...

overview

This page introduces "logarithm".

One of the inverses of exponent is logarithm. Logarithm is introduced with the following two.

•  first principles -- Logarithm of a number is the power in the equivalent exponent.

•  Simplified Procedure -- Logarithm of a number is found from prime-factorization of the number (if log evaluates to an integer).

base to the power

We learned that
•  Subtraction is the inverse of addition,
•  Division is the inverse of multiplication,

Similarly, the "exponent" has two inverses

given result of exponentiation and power, find the base

given result of exponentiation and base, find the power

The exponent is not commutative and that results in two inverses for exponents.
${3}^{2}=9$${3}^{2} = 9$ and ${2}^{3}=8$${2}^{3} = 8$
${a}^{b}\ne {b}^{a}$${a}^{b} \ne {b}^{a}$

Exponent is
${\left(\text{base}\right)}^{\text{power}}=\text{exp. result}$${\left(\textrm{b a s e}\right)}^{\textrm{p o w e r}} = \textrm{\exp . r e s \underline{t}}$

If exponentiation result and power are given, then
$\text{base}=\sqrt[\text{power}]{\text{result}}={\left(\text{exp. result}\right)}^{\frac{1}{\text{power}}}$$\textrm{b a s e} = \sqrt[\textrm{p o w e r}]{\textrm{r e s \underline{t}}} = {\left(\textrm{\exp . r e s \underline{t}}\right)}^{\frac{1}{\textrm{p o w e r}}}$

This inverse is called "root"
.

If exponentiation-result and base are given, then
$\text{power}={\mathrm{log}}_{\text{base}}\left(\text{exp.result}\right)$$\textrm{p o w e r} = {\log}_{\textrm{b a s e}} \left(\textrm{\exp . r e s \underline{t}}\right)$
This inverse is called "logarithm"
.

The word "logarithm" means: numbers in ratio order.

The word logarithm is derived from original Greek words, "logos"( meaning ratio) and "arithmos" (meaning numbers).

logarithm is associated with a sequence increasing or decreasing in ratios, like $1,10,100,\cdots$$1 , 10 , 100 , \cdots$.

Logarithm : Logarithm of a number to a given base of logarithm, is the power of the base that equals the number.

${\mathrm{log}}_{2}8=3$${\log}_{2} 8 = 3$
$2$$2$ is the base of logarithm
$8$$8$ is the number for which logarithm is calculated
$3$$3$ is the result of logarithm

${\mathrm{log}}_{2}8=3$${\log}_{2} 8 = 3$ implies that ${2}^{3}=8$${2}^{3} = 8$, and the operation logarithm finds the power in the exponent.

Finding Logarithm (First Principles) : Logarithm of a number is the power in the equivalent exponent.

eg: ${\mathrm{log}}_{4}64$${\log}_{4} 64$ is seen as exponent $64={4}^{3}$$64 = {4}^{3}$. The power is $3$$3$ and so ${\mathrm{log}}_{4}64=3$${\log}_{4} 64 = 3$

"${\mathrm{log}}_{7}49$${\log}_{7} 49$" is to find the power from ${7}^{2}=49$${7}^{2} = 49$

Consider ${\mathrm{log}}_{5}\left(125\right)$${\log}_{5} \left(125\right)$
To find ${\mathrm{log}}_{5}\left(125\right)$${\log}_{5} \left(125\right)$, represent the value in the given base.
$125=5×5×5={5}^{3}$$125 = 5 \times 5 \times 5 = {5}^{3}$

By first principles, ${\mathrm{log}}_{5}\left(125\right)=3$${\log}_{5} \left(125\right) = 3$

To find ${\mathrm{log}}_{6}\left(36\right)$${\log}_{6} \left(36\right)$, represent the value in the given base.
$36=6×6={6}^{2}$$36 = 6 \times 6 = {6}^{2}$

By first principles, ${\mathrm{log}}_{6}\left(36\right)=2$${\log}_{6} \left(36\right) = 2$

Finding Logarithms (Simplified Procedure) : To find logarithm of a number, express the number in the given base.

eg: ${\mathrm{log}}_{10}1000$${\log}_{10} 1000$ $={\mathrm{log}}_{10}\left(10×10×10\right)$$= {\log}_{10} \left(10 \times 10 \times 10\right)$ $=3$$= 3$

summary

Logarithm : Logarithm of a number to a given base of logarithm, is the power of the base that equals the number.

${\mathrm{log}}_{2}8=3$${\log}_{2} 8 = 3$
$2$$2$ is the base of logarithm
$8$$8$ is the number for which logarithm is calculated
$3$$3$ is the result of logarithm

${\mathrm{log}}_{2}8=3$${\log}_{2} 8 = 3$ implies that ${2}^{3}=8$${2}^{3} = 8$, and the operation logarithm finds the power in the exponent.

Finding Logarithm (First Principles) : Logarithm of a number is the power in the equivalent exponent.

eg: ${\mathrm{log}}_{4}64$${\log}_{4} 64$ is seen as exponent $64={4}^{3}$$64 = {4}^{3}$. The power is $3$$3$ and so ${\mathrm{log}}_{4}64=3$${\log}_{4} 64 = 3$

Finding Logarithms (Simplified Procedure) : To find logarithm of a number, express the number in the given base.

eg: ${\mathrm{log}}_{10}1000$${\log}_{10} 1000$ $={\mathrm{log}}_{10}\left(10×10×10\right)$$= {\log}_{10} \left(10 \times 10 \times 10\right)$ $=3$$= 3$

Outline

The outline of material to learn "Exponents" is as follows. Note: click here for detailed outline of Exponents s

→   Representation of Exponents

→   Inverse of exponent : root

→   Inverse of exponent : Logarithm

→   Common and Natural Logarithms

→   Exponents Arithmetics

→   Logarithm Arithmetics

→   Formulas

→   Numerical Expressions

→   PEMA / BOMA

→   Squares and Square roots

→   Cubes and Cube roots