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. ${3}^{2}=9$ and ${2}^{3}=8$ $a}^{b}\ne {b}^{a$ *

Exponent is

$\left(\text{base}\right)}^{\text{power}}=\text{exp. result$

*If exponentiation result and power are given, then
$\text{base}=\sqrt[\text{power}]{\text{result}}={\left(\text{exp. result}\right)}^{\frac{1}{\text{power}}}$
This inverse is called "root"*.

$\text{power}={\mathrm{log}}_{\text{base}}\left(\text{exp.result}\right)$

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 $1,10,100,\cdots$.

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

${\mathrm{log}}_{2}8=3$

$2$ is the base of logarithm

$8$ is the number for which logarithm is calculated

$3$ is the result of logarithm

${\mathrm{log}}_{2}8=3$ implies that ${2}^{3}=8$, 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: ${\mathrm{log}}_{4}64$ is seen as exponent $64={4}^{3}$. The power is $3$ and so ${\mathrm{log}}_{4}64=3$

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

Consider ${\mathrm{log}}_{5}\left(125\right)$

To find ${\mathrm{log}}_{5}\left(125\right)$, represent the value in the given base.

$125=5\times 5\times 5={5}^{3}$

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

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

$36=6\times 6={6}^{2}$

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

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

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

summary

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

${\mathrm{log}}_{2}8=3$

$2$ is the base of logarithm

$8$ is the number for which logarithm is calculated

$3$ is the result of logarithm

${\mathrm{log}}_{2}8=3$ implies that ${2}^{3}=8$, 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: ${\mathrm{log}}_{4}64$ is seen as exponent $64={4}^{3}$. The power is $3$ and so ${\mathrm{log}}_{4}64=3$

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

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

Outline

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

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