# How do you do logarithms in math

## Introduction to 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 had to multiply 3 of the 2s to get 8

So the logarithm is 3

### How to Write it

We write "the number of 2s we need to multiply to get 8 is 3" as:

log2(8) = 3

So these two things are the same:

The number we multiply is called the "base", so we can say:

• "the logarithm of 8 with base 2 is 3"
• or "log base 2 of 8 is 3"
• or "the base-2 log of 8 is 3"

### Notice we are dealing with three numbers:

• the base: the number we are multiplying (a "2" in the example above)
• how often to use it in a multiplication (3 times, which is the logarithm)
• The number we want to get (an "8")

## More Examples

### Example: What is log5(625) ... ?

We are asking "how many 5s need to be multiplied together to get 625?"

5 × 5 × 5 × 5 = 625, so we need 4 of the 5s

Answer: log5(625) = 4

### Example: What is log2(64) ... ?

We are asking "how many 2s need to be multiplied together to get 64?"

2 × 2 × 2 × 2 × 2 × 2 = 64, so we need 6 of the 2s

Answer: log2(64) = 6

## Exponents

Exponents and Logarithms are related, let's find out how ...

 The exponent says how many times to use the number in a multiplication.In this example: 23 = 2 × 2 × 2 = 8(2 is used 3 times in a multiplication to get 8)

So a logarithm answers a question like this:

In this way:

The logarithm tells us what the exponent is!

In that example the "base" is 2 and the "exponent" is 3:

So the logarithm answers the question:

What exponent do we need
(for one number to become another number)
?

The general case is:

Example: What is log10(100) ... ?

102 = 100

So an exponent of 2 is needed to make 10 into 100, and:

log10(100) = 2

Example: What is log3(81) ... ?

34 = 81

So an exponent of 4 is needed to make 3 into 81, and:

log3(81) = 4

## Common Logarithms: Base 10

Sometimes a logarithm is written without a base, like this:

log(100)

This usually means that the base is really 10.

It is called a "common logarithm". Engineers love to use it.

On a calculator it is the "log" button.

It is how many times we need to use 10 in a multiplication, to get our desired number.

Example: log(1000) = log10(1000) = 3

## Natural Logarithms: Base "e"

Another base that is often used is e (Euler's Number) which is about 2.71828.

This is called a "natural logarithm". Mathematicians use this one a lot.

On a calculator it is the "ln" button.

It is how many times we need to use "e" in a multiplication, to get our desired number.

Example: ln(7.389) = loge(7.389) ≈ 2

Because 2.718282 ≈ 7.389

## But Sometimes There Is Confusion ... !

Mathematicians use "log" (instead of "ln") to mean the natural logarithm. This can lead to confusion:

ExampleEngineer
Thinks
Mathematician
Thinks

log(50)log10(50)loge(50)confusion
ln(50)loge(50)loge(50)no confusion
log10(50)log10(50)log10(50)no confusion

So, be careful when you read "log" that you know what base they mean!

## Logarithms Can Have Decimals

All of our examples have used whole number logarithms (like 2 or 3), but logarithms can have decimal values like 2.5, or 6.081, etc.

Example: what is log10(26) ... ?

 Get your calculator, type in 26 and press logAnswer is: 1.41497...

The logarithm is saying that 101.41497... = 26
(10 with an exponent of 1.41497... equals 26)

 This is what it looks like on a graph:See how nice and smooth the line is.

Read Logarithms Can Have Decimals to find out more.

## Negative Logarithms

 − Negative? But logarithms deal with multiplying. What is the opposite of multiplying? Dividing!

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

We can have just one divide:

Example: What is log8(0.125) ... ?

Well, 1 ÷ 8 = 0.125,

So log8(0.125) = −1

Or many divides:

Example: What is log5(0.008) ... ?

1 ÷ 5 ÷ 5 ÷ 5 = 5−3,

So log5(0.008) = −3

## It All Makes Sense

Multiplying and Dividing are all part of the same simple pattern.

Let us look at some Base-10 logarithms as an example:

 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..

Looking at that table, see how positive, zero or negative logarithms are really part of the same (fairly simple) pattern.

## 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" !

Exponents, Roots and LogarithmsWorking with Exponents and LogarithmsAlgebra Index