# Logarithms

Logarithms, or “logs,” are a part of mathematics. They are related to exponential functions, and are useful in multiplying and dividing large numbers.

## Contents

# Relationship with exponential functions

A logarithm tells what exponent (or power) is needed to make a certain number. Thus, logarithms are the inverse (the opposite) of exponentiation^{[1]}. That is, what an exponent does, the logarithm “undoes.”

Exponential functions have three parts, and logarithms have basically the same three parts. The three parts of an exponential function are:

- the
**base** - the number that we are multiplying by itself.
- the
**argument** - the number of times that we multiply the base. The
*argument*is often called the*power*or the*exponent*. - the
**answer** - the number we get when we multiply the base by itself the argument number of times. So, you know, the answer.

The three parts of a logarithm are:

- the
**base** - the number that we are multiplying by itself.
*This is the same as the base of the exponential function.* - the
**argument** - the number we get when we multiply the base by itself the answer number of times.
*This is the the answer from the exponential function.* - the
**answer** - the number of times that we multiply the base to get the argument.
*This is the argument from the exponential function.*

Here is an example of an **exponential** function:

**<math>2^{3}=8</math>**

In this function, the base is <math>2</math>, the argument is <math>3</math> and the answer is <math>8</math>. This is easier to see when you remember that <math>2^{3}=8</math> can also be written as <math>2\cdot{}2\cdot{}2=8</math>.

The logarithm of the above exponential function is just the “opposite”, or **inverse**:

**<math>\log_2(8)=3</math>**

In this logarithm, the base is still <math>2</math>, the argument this time is <math>8</math>, and the answer is <math>3</math>. Notice that the base did not change, but the argument and answer switched. This is because the exponential function and logarithm are inverse—they “undo” each other. The *answer* of an exponential function is the *argument* of the logarithmic function.

So, you might understand the above equation as saying “To get <math>8</math>, I need to multiply <math>2</math> by itself <math>3</math> times.”

Here is another example:

**<math>\log_{10}(400) \approx 2.6021</math>**

This is saying, “To get <math>400</math>, we need to multiply <math>10</math> by itself approximately <math>2.6021</math> times.^{[2]} We can show this exponentially as <math>10^{2.6021} \approx{} 400</math>.

Notice that you can have non-whole number answers to logarithms. Also, as the above example shows, any whole number can be written as a base raised to a real-number exponent.

## Converting to and from logarithmic functions

Because exponential and logarithmic functions include the same information (the *base*, the *argument*, and the *answer*), they can easily be **converted** from one to the other. We saw this in the earlier example: <math>2^{3}=8</math> can be converted into <math>\log_2(8)=3</math>. This is useful when one of the three parts is a variable.

For example:

**<math>\log_2(x)=3</math>**

We can find <math>x</math> by converting the logarithm into exponential form:

**<math>2^{3}=x</math>**

We can easily solve this to find that **<math>x=8</math>**.

## Isolating logarithms

When a variable appears as the *base*, the *argument*, or the *answer*, you may need to **isolate** the logarithm. This means that you get the equation into proper logarithmic form so that you can **convert** it to exponential form.

Basically, **isolating** the logarithm is similar to what you do when you solve for <math>y</math> in a simple algebraic equation. The difference is just that you are getting the logarithm by itself (thus, *isolating* it) on one side of the equation.

For example, if you have this equation:

**<math>\log_{10}(x-3)+4=3</math>**

you would have to <math>-4</math> from both sides to isolate the <math>\log_{10}(x-3)</math> before you could solve the logarithm part.

[[File:./john_napier.jpg|frame|none|alt=|caption The spiral of a shell is logarithmic.]]

[[File:./NautilusCutawayLogarithmicSpiral.jpg|frame|none|alt=|caption The spiral of a shell is logarithmic.]]

# Common Logarithms

Logarithms to base 10 are called common logarithms. They are usually written without the base. For example: <math>\log(100)=2</math>, which means <math>10^{2}=100</math>.

# Natural Logarithms

Logarithms to base <math>e</math> are called natural logarithms. The number <math>e</math> is nearly 2.71828, and is also called the Eulerian constant after the mathematician Leonhard Euler.

The natural logarithms can take the symbols <math>\log_{e}(x)</math> or <math>\ln(x)</math>.

# Common bases for logarithms

r|c|l Base&Abbreviation&Comment

2&<math>\operatorname {ld}</math>&Very common in Computer Science (binary)

<math>e</math>&<math>\ln{}</math>&The base of this is the Eulerian constant <math>e</math>.

10&<math>\lg{}</math> often <math>\log{}</math>&Most people use base 10.

any number, <math>n</math>&<math>\log_n</math>&This is the general way to write logarithms

# Properties of logarithms

Logarithms have many properties.

This property is straight from the definition of a logarithm:

<math>\log_n(n^{a})=a</math>

Examples:

<math>\log_2(2^{3})=3</math>.

<math>\log_2\frac{1}{2}=-1</math> because <math>\frac{1}{2}=2^{-1}</math>.

The logarithm to base <math>b</math> of a number <math>a</math> is the same as the logarithm of <math>a</math> divided by the logarithm of <math>b</math>. That is,

<math>\log_b(a)=\frac{\log(a)}{\log(b)}</math>

For example, let <math>a</math> be <math>6</math> and <math>b</math> be <math>2</math>. With calculators we can show that this is true or at least very close:

<math>\log_2(6)=\frac{\log(6)}{\log(2)}</math>.

<math>\log_2(6) \approx 2.584962</math>.

<math>2.584962 \approx \frac{0.778151}{0.301029} \approx 2.584970</math>

The result has a small error, but this was due to the rounding of numbers.

The important thing to notice is the switch from *log* base <math>2</math> to *log* base <math>10</math>. What that example says is that the *log* of <math>6</math> in base <math>2</math> is equal to the *log* of <math>6</math> in base <math>10</math> divided by the *log* of <math>2</math> in base <math>10</math>. This property of logarithms—changing bases—can be very useful.

Because it is hard to picture the natural logarithm, we find that, in terms of a base-ten logarithm:

<math>\ln(x)=\frac{\log(x)}{\log(e)}\approx \frac{\log(x)}{0.434294}</math>

Where 0.434294 is an approximation for the logarithm of <math>e</math>.

## Operations within logarithm arguments

Logarithms which multiply inside their argument can be changed as follows:

<math>\log(ab)=\log(a)+\log(b)</math>

For example,

<math>\log(1000)=\log(10\cdot 10\cdot 10)=\log(10)+\log(10)+\log(10)=1+1+1=3</math>

The same thing works for dividing, but using subtraction instead of addition, because division is the inverse operation of multiplication:

<math>\log(\frac{a}{b})=\log(a)-\log(b)</math>

# Logarithms and Roots

Addition has one inverse operation: subtraction. Multiplication has one inverse operation: division. But, exponentiation has **two** inverse operations! This is because exponentiation is not *commutative*.^{[3]}

The following examples show this:

- If you have <math>x+2=3</math>,
- then you can use subtraction to find out that <math>x=3-2</math>. This is the same if you have <math>2+x=3</math>: you also get <math>x=3-2</math>. This is because <math>x+2</math> is the same as <math>2+x</math>.

<math>x+y=y+x</math>. *Addition is commutative*.

- If you have <math>x\cdot{}2=3</math>,
- then you can use division to find out that <math>x=\frac{3}{2}</math>. This is the same if you have <math>2\cdot{}x=3</math>: you also get <math>x=\frac{3}{2}</math>. This is because <math>x\cdot{}2</math> is the same as <math>2\cdot{}x</math>.

<math>x\cdot{}y=y\cdot{}x</math>. *Multiplication is commutative*.

- If you have <math>x^{2}=3</math>,
- then you use the square root to find out <math>x</math>: you get the result <math>x=\sqrt{3}</math>. But, if you have <math>2^{x}=3</math>, then you
*cannot*use the root to find out <math>x</math>. Instead, you have to use the logarithm to find out <math>x</math>: you get the result <math>x=log_{2}(3)</math>. This is because <math>2^{x}</math> is not the same as <math>x^{2}</math> (except in the special case that <math>x=2</math>).

<math>y^{x}\neq{}x^{y}</math> (unless <math>x=y</math>).

*Using exponents is (usually) not commutative*.

# Uses

There are many, many uses for logarithms. Here are a few from nature and a few from human activities:

- Logarithmic spirals are common in nature. Examples include the shell of a nautilus or the arrangement of seeds on a sunflower.
- Musical intervals are measured logarithmically.
- The Richter scale measures earthquake intensity on a base-10 logarithmic scale.
- In astronomy, the apparent magnitude of stars (how bright they appear) is measured logarithmically. This is because the human eye responds logarithmically to brightness.
- Logarithms can make multiplication and division of large numbers easier because adding logarithms is the same as multiplying, and subtracting logarithms is the same as dividing.
- Before calculators became cheap and common, people used logarithm tables in books to multiply and divide. The same information in a logarithm table was available on a slide rule, a tool with logarithms written on it.

[[File:./slide_rule.jpg|frame|none|alt=|caption A slide rule with cursor]]

[[File:./slide_rule_cursor.jpg|frame|none|alt=|caption A slide rule with cursor]]

# Sources

Modified from the content available at the following pages:

- https://simple.wikipedia.org/wiki/Logarithm
- https://simple.wikipedia.org/wiki/File:NautilusCutawayLogarithmicSpiral.jpg (Thanks to Chris 73 and Wikimedia Commons for the picture)
- https://simple.wikipedia.org/wiki/File:John_Napier.jpg (Public Domain)
- https://simple.wikipedia.org/wiki/File:Sliderule.PickettN902T.agr.jpg
- https://simple.wikipedia.org/wiki/File:Slide_rule_cursor.jpg

# Copyright

This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.

The documents and images used to create this W5H Reading are available at

https://www.osugisakae.com/tesol/teaching-materials/w5h/math/logarithms/.