Logarithms

 In its simplest form, a logarithm answers the question:

How many of one number do we multiply to get another number?

Example

How many 2s do we multiply to get 8?

Answer: 2 × 2 × 2 = 8, so we needed to multiply 3 of the 2s to get 8

So the logarithm is 3

So these two things are the same:

 

 

 

Negative Logarithms

A negative logarithm means how many times to divide by the number. 

Example: What is log₅(0.008) ... ?
1 ÷ 5 ÷ 5 ÷ 5 = 5^-3, so log₅(0.008) = -3

	Number	How Many 10s	Base-10 Logarithm
	.. etc..
	1000	1 × 10 × 10 × 10	log10(1000)	=	3
	100	1 × 10 × 10	log10(100)	=	2
	10	1 × 10	log10(10)	=	1
	1	1	log10(1)	=	0
	0.1	1 ÷ 10	log10(0.1)	=	-1
	0.01	1 ÷ 10 ÷ 10	log10(0.01)	=	-2
	0.001	1 ÷ 10 ÷ 10 ÷ 10	log10(0.001)	=	-3
	.. etc..

The Word

"Logarithm" is a word made up by Scottish mathematician John Napier (1550-1617), from the Greek word logos meaning "proportion, ratio or word" and arithmos meaning "number", ... which together makes "ratio-number" !

—The Relationship Animated—

 

The graph of the logarithm function log_b(x) (blue)

 is obtained by reflecting  the graph of the 

function b^x (red) at the diagonal line (x = y).

 

The graph of the natural logarithm (green) and 

its tangent at x = 1.5 (black) 

The natural logarithm of t agrees with the integral of 1/x dx from 1 to t:

 

The natural logarithm of t is the shaded area 

underneath the graph of the function

 f(x) = 1/x (reciprocal of x).