LOGARITHMS

WHAT IS LOGARITHM?

Introduction

A logarithm is the power to which a number must be raised in order to get some other number.

For example, the base ten logarithm of 100 is 2, because ten raised to the power of two is 100:

log 100 = 2

because

10² = 100

This is an example of a base-ten logarithm. We call it a base ten logarithm because ten is the number that is raised to a power. The base unit is the number being raised to a power. There are logarithms using different base units. If you wanted, you could use two as a base unit. 

For instance, the base two logarithm of eight is three, because two raised to the power of three equals eight: 

log₂ 8 = 3

because

2³ = 8 

Law of Logarithm

Example

What is a logarithm?

Logarithms are another way of thinking about exponents.

Both equations describe the same relationship between the numbers $\blueD2$start color blueD, 2, end color blueD, $\greenE4$start color greenE, 4, end color greenE, and$\goldD{16}$start color goldD, 16, end color goldD, where $\blueD2$start color blueD, 2, end color blueD is the base, $\greenE4$start color greenE, 4, end color greenE is the exponent, and $\goldD{16}$start color goldD, 16, end color goldD is the power.

$ex.\; 2$⁴$=$$16$

start color blueD, 2, end color blueD, start superscript, start color greenE, 4, end color greenE, end superscript, equals, start color goldD, 16, end color goldD

24=16⟺log2(16)=4.

The difference is that while the exponential form isolates the power, 

$\goldD{16}$start color goldD, 16, end color goldD, the logarithmic form isolates the exponent, $\greenD 4$start color greenD, 4, end color greenD.

Here are more examples of equivalent logarithmic and exponential equations.

Logarithmic form		Exponential form
$\log_\blueD2(\goldD{8})=\greenD3$log, start subscript, start color blueD, 2, end color blueD, end subscript, left parenthesis, start color goldD, 8, end color goldD, right parenthesis, equals, start color greenD, 3, end color greenD	⟺	$\blueD2^\greenD3=\goldD8$start color blueD, 2, end color blueD, start superscript, start color greenD, 3, end color greenD, end superscript, equals, start color goldD, 8, end color goldD
$\log_\blueD3(\goldD{81})=\greenD4$log, start subscript, start color blueD, 3, end color blueD, end subscript, left parenthesis, start color goldD, 81, end color goldD, right parenthesis, equals, start color greenD, 4, end color greenD	⟺	$\blueD3^\greenD4=\goldD{81}$start color blueD, 3, end color blueD, start superscript, start color greenD, 4, end color greenD, end superscript, equals, start color goldD, 81, end color goldD
$\log_\blueD5(\goldD{25})=\greenD2$log, start subscript, start color blueD, 5, end color blueD, end subscript, left parenthesis, start color goldD, 25, end color goldD, right parenthesis, equals, start color greenD, 2, end color greenD	⟺	$\blueD5^\greenD2=\goldD{25}$start color blueD, 5, end color blueD, start superscript, start color greenD, 2, end color greenD, end superscript, equals, start color goldD, 25, end color goldD

Definition of a logarithm

Generalizing the examples above leads us to the formal definition of a logarithm.

logb(a)=c⟺bc=a

Both equations describe the same relationship between $\goldD a$start color goldD, a, end color goldD, $\blueD b$start color blueD, b, end color blueD, and $\greenE c$start color greenE, c, end color greenE:

• $\blueD b$start color blueD, b, end color blueD is the $\blueD{\text{base}}$start color blueD, b, a, s, e, end color blueD,
• $\greenE c$start color greenE, c, end color greenE is the $\greenE{\text{exponent}}$start color greenE, e, x, p, o, n, e, n, t, end color greenE, and
• $\goldD a$start color goldD, a, end color goldD is the $\goldD{\text{power}}$start color goldD, p, o, w, e, r, end color goldD.

In logarithm form, this is also called the $\goldD{\text{argument}}$start color goldD, a, r, g, u, m, e, n, t, end color goldD.

References:

https://www.khanacademy.org/math/algebra2/exponential-and-logarithmic-functions/introduction-to-logarithms/a/intro-to-logarithms

https://www.youtube.com/watch?v=Z5myJ8dg_rM