# An Introduction to Logarithms: How to Use the Rules of Logs and Exponents

## An Introduction to Logarithms, Bases and Exponents

In this tutorial you'll learn about

- exponentiation
- bases
- logarithms to the base 10
- natural logarithms
- rules of exponents and logarithms
- working out logarithms on a calculator
- graphs of logarithmic functions
- the uses of logarithms
- using logarithms to perform multiplication and division

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## What is Exponentiation?

Before we learn about logarithms, we need to understand the concept of exponentiation. Exponentiation is a math operation that raises a number to a power of another number to get a new number.

So 10^{2 }= 10 x 10 = 100

Similarly 4^{3} = 4 x 4 x 4 = 64

and 25^{5} = 2 x 2 x 2 x 2 x 2 = 32

We can also raise numbers with decimal parts (non-integers) to a power.

So 1.5^{2 }= 1.5 x 1.5 = 2.25

## What are Bases and Exponents?

In general, if b is an integer:

Then a

^{b}= a x a x a x a....a = c

a is called the base and b is called the exponent. As we'll find out later, b doesn't have to be an integer and can be a decimal.

## How to Simplify Expressions Involving Exponents

There are several laws of exponents (sometimes called "rules of exponents") we can use to simplify expressions that include numbers or variables raised to a power.

## Laws of Exponents

## Examples Using the Laws of Exponents

### Zero exponent

5^{0} = 1

27^{0 }= 1

1000^{0 }= 1

### Negative exponent

2^{-4} = 1/2^{4} = 1/16

10^{-3} = 1/10^{3} = 1/1000^{}

### Product law

5^{2} x 5^{3} = 5 ^{(2 + 3) }= 5^{5} = 3125

### Quotient law

3^{4 }/ 3^{2} = 3 ^{(4 - 2) }= 3^{2} = 9

### Power of a power

(2^{3})^{4} = 2^{12} = 4096

### Power of a product

(2 x 3)^{2} = 6^{2 }= 36 = (2^{2} x 3^{2}) = 4 x 9 = 36

## Exercise A: Laws of Exponents

Simplify the following:

- y
^{a}y^{b}y^{c} - p
^{a}p^{b}/p^{x}p^{y} - p
^{a}p^{b}/q^{x}q^{y} - ((ab)
^{4})^{3 }x ((ab)^{2})^{3} - ( ((ab)
^{4})^{3 }x ((ab)^{4})^{3 })^{2}/ a^{25}

Answers at bottom of page.

## Non-Integer Exponents

Exponents don't have to be integers, they can also be decimals.

For instance imagine if we have a number b, then the product of the square roots of b is b

So √b x √b = b

Now instead of writing √b we write it as b raised to a power x:

Then √b = b^{x }and b^{x} x b^{x }= b

But using the product rule and the quotient of one rule we can write:

b^{x} x b^{x }= b^{2x }= b = b^{1}

So b^{2x} = b^{1} Therefore 2x = 1 and x = 1/2

So √b = b^{x }= b^{1/2}

## What are Logarithms?

If we raise 10 to the power of 3, we get 1000.

10^{3 = }10 x 10 x 10 = 1000

The logarithm function is the reverse of exponentiation and the logarithm of a number (or *log *for short) is the number a base must be raised to, to get that number.

So log_{10} 1000 = 3 because 10 must be raised to the power of 3 to get 1000.

We indicate the base with the subscript _{10} in log_{10 . }Sometimes this is omitted.

### Some more examples:

10^{2} = 100 and log_{10}100 = 2

10^{4 }= 10,000 and log_{10} 10,000 = 4

10^{6} = 1000,000 and log_{10} 1000,000 = 6

The logarithm of a number (or log for short) is the number a base must be raised to, to get that number.

## Logarithms to Bases Other than 10

We can of course work out logs to other bases.

### Some examples:

2^{3} = 8 and log_{2} 8 = 3

3^{4} = 81 and log_{3} 81 = 4

## How to Work Out Logarithms Using a Calculator

You can use the log function on a calculator to work out the log of a number to the base 10.

- Press "log".
- Type the number.
- You may have to press "=" depending on the model of the calculator.

To work out the log of a number to a base other than 10:

- Press the "2nd function" or "shift" key
- Press the "log
_{y }x" key - Type the base
- Type the number
- You may have to press "=" depending on the model of the calculator.

Not all calculators have a "log_{y }x" key, so see "change of base" in properties of logarithms below.

## Exercise B: Calculate Values of Logs

Calculate the value of these logs:

- log
_{2}256 - log
_{10}1000,000 - log
_{3}81 - log
_{1.5}3.375 - ln 20.0855

Answers at bottom of page

## The Natural Logarithm

The mathematical constant *e *known as* Euler's number *is* *approximately equal to 2.71828

The value of the expression (1 + 1/n)^{n} approaches e as n gets bigger and bigger.

The derivative of e^{x }is itself. So d/dx(e^{x}) = e^{x}

The log of a number x to the base e is normally written as ln *x* or log_{e}*x*

## Graph of the Log Function

The graph below shows the function log (x) for the bases 10, 2 and e.

We notice several properties about the log function:

- Since x
^{0}= 1 for all values of x, log (1) for all bases is 0. - Log x increases at a decreasing rate as x increases.
- Log 0 is undefined. Log x tends to -∞ as x tends towards 0.

## Properties of Logarithms

These are sometimes called logarithmic identities or logarithmic laws.

### The product rule:

The log of a product equals the sum of the logs.

log_{c }(*AB*) = log_{c }*A* + log_{c }*B*

### The quotient rule:

The log of a quotient (i.e. a ratio) is the difference between the log of the numerator and the log of the denominator.

log_{c }(*A/B*) = log_{c }*A* - log_{c }*B*

### The power rule:

The log of a number raised to a power is the product of the power and the number.

log_{c} (*A ^{b}*) =

*b*log

_{c}

*A*

### Change of base:

log_{c }*A* = log_{b}* A /* log_{b}*c*

This identity is useful if you need to work out a log to a base other than 10. Many calculators only have "log" and "ln" keys for log to the base 10 and natural log to the base *e* respectively.

### Example

What is log_{2}256 ?

log_{2}256 = log_{10}256 / log_{10}2 = 8

## Exercise C: Using Rules of Logs to Simplify Expressions

Simplify the following:

- log
_{10}35x - log
_{10}5/x - log
_{10}x^{5} - log
_{10 }10x^{3} - log
_{2}8x^{4} - log
_{3}27(x^{2}/y^{4}) - log
_{5}(1000) in terms of the base 10, rounded to two decimal places

## What are Logarithms Used For?

- Representing numbers with a large dynamic range
- Compressing scales on graphs
- Multiplying and dividing decimals
- Simplifying functions to work out derivatives

## Representing Numbers With a Large Dynamic Range

In science, measurements can have a large dynamic range. This means that there can be a huge variation between the smallest and largest value of a parameter.

### Sound pressure levels

An example of a parameter with a large dynamic range is sound.

Typically sound pressure level (SPL) measurements are expressed in decibels.

Sound pressure level = 20log_{10 }( *p / p _{0}*)

where *p* is the pressure and *p _{o }*is a reference pressure level (20 μPa, the faintest sound the human ear can hear)

By using logs, we can represent levels from 20 μPa = 20 x 10^{-5} Pa up to the sound level of a rifle gunshot (7265 Pa) or higher on a more usable scale of 0dB to 171dB.

So if p is 20 x 10^{-5}, the faintest sound we can hear

Then SPL = 20log_{10 }( *p / p _{0}*)

= 20log_{10 }( 20 x 10^{-5}* / 20 x 10 ^{-5}*)

= 20log_{10 }(1) = 20 x 0 = 0dB

If sound is 10 times louder , i.e. 20 x 10^{-4}

Then SPL = 20log_{10 }( *p / p _{0}*)

= 20log_{10 }( 20 x 10^{-4}* / 20 x 10 ^{-5}*)

= 20log_{10 }(10) = 20 x 1 = 20dB

Now increase the sound level by another factor of 10, i.e. make it 100 times louder than the faintest sound we can hear.

So p = 20 x 10^{-3}

SPL = 20log_{10 }( *p / p _{0}*)

= 20log_{10 }( 20 x 10^{-3}* / 20 x 10 ^{-5}*)

= 20log_{10 }(100) = 20 x 2 = 40dB

So each 20DB increase in SPL represents a tenfold increase in level of sound pressure.

### Richter magnitude scale

The magnitude of an earthquake on the Richter scale is determined by using a seismograph to measure the amplitude of ground movement waves. The log of the ratio of this amplitude to a reference level gives the strength of the earthquake on the scale.

The original scale is log_{10 }( A / A_{0} ) where A is the amplitude and A_{0} is the reference level. Similar to sound pressure measurements on a log scale, every time the value on the scale increases by 1, this represents a tenfold increase in strength of the earthquake. So an earthquake of strength 6 on the Richter scale is ten times stronger than a level 5 earthquake and 100 times stronger than a level 4 quake.

## Logarithmic Scales on Graphs

Values with a large dynamic range are often represented on graphs with nonlinear, logarithmic scales. The x-axis or y-axis or both can be logarithmic, depending on the nature of data represented. Each division on the scale normally represents a tenfold increase in value. Typical data displayed on a graph with a logarithmic scale is:

- Sound pressure level (SPL)
- Sound frequency
- Earthquake magnitudes (Richter scale)
- pH (acidity of a solution)
- Light intensity
- Tripping current for circuit breakers and fuses

## Answers to Exercises

**Exercise A**

- y
^{(a + b +c)} - p
^{(a + b -x - y)} - p
^{(a +}^{b)}/q^{(x + y)} - (ab)
^{18} - a
^{23}b^{48}

**Exercise B**

- 8
- 6
- 4
- 3
- 3

**Exercise C**

- log
_{10}35 + log_{10 }x - log
_{10}5 - log_{10}x - 5log
_{10}x - 1 + 3log
_{10}x - 3 + 4log
_{2}x - 3 + 2log
_{3}x - 4log_{3}y - log
_{10}1000 / log_{10}5 = 4.29 approx

**© 2019 Eugene Brennan**

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