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Rules of Logarithms, Bases, and Exponents: Lessons and Exercises

Eugene is a qualified control/instrumentation engineer Bsc (Eng) and has worked as a developer of electronics & software for SCADA systems.

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
  • the history and uses of logarithms
  • working out logarithms on a calculator
  • graphs of logarithmic functions

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 102 = 10 x 10 = 100

Similarly 43 = 4 x 4 x 4 = 64

and 25 = 2 x 2 x 2 x 2 x 2 = 32

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

So 1.52 = 1.5 x 1.5 = 2.25

What are Bases and Exponents?

In general, if b is an integer:

Then ab = 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 rational number with a decimal part or even an irrational number such as the square root of two.

an-introduction-to-logarithms

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

Laws of exponents (rules of exponents).

Laws of exponents (rules of exponents).

Examples Using the Laws of Exponents

Zero exponent

50 = 1
270 = 1
10000 = 1

Exponent of 1

51 = 5

20001 = 2000

3.251 = 3.25

Negative exponent

2-4 = 1/24 = 1/16

10-3 = 1/103 = 1/1000

Product law

52 x 53 = 5 (2 + 3) = 55 = 3125

Quotient law

34 / 32 = 3 (4 - 2) = 32 = 9

Power of a power

(23)4 = 212 = 4096

Power of a product

(2 x 3)2 = 62 = 36 = (22 x 32) = 4 x 9 = 36

Exercise A: Laws of Exponents

Simplify the following:

  1. yaybyc
  2. papb/pxpy
  3. papb/qxqy
  4. ((ab)4)3 x ((ab)2)3
  5. ( ((ab)4)3 x ((ab)4)3 )2 / a25

Answers at bottom of page.

Non-Integer Exponents

Exponents don't have to be integers, they can also be rational numbers with decimal parts or irrational numbers such as √2.

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 = bx and bx x bx = b

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

bx x bx = b2x = b = b1

So b2x = b1 Therefore 2x = 1 and x = 1/2

So √b = bx = b1/2

What are Logarithms?

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

103 = 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 log10 1000 = 3 because 10 must be raised to the power of 3 to get 1000.

We indicate the base with a subscript, the number 10 in the case of log to the base 10. So for example the log of 5 to the base 10 is written as:

log10 5

In the specific case of log to the base 10, the subscript 10 is often omitted.

Some more examples:

102 = 100 and log10100 = 2

104 = 10,000 and log10 10,000 = 4

106 = 1000,000 and log10 1000,000 = 6

So in general

if c = ab

then

log c = b

Log to the base 10.

Log to the base 10.

Logarithms to Bases Other than 10

We can of course work out logs to other bases.

Some examples:

23 = 8 and log2 8 = 3

34 = 81 and log3 81 = 4

Log to the base 2.

Log to the base 2.

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.

  1. Press "log".
  2. Type the number.
  3. 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:

  1. Press the "2nd function" or "shift" key
  2. Press the "logy x" key
  3. Type the base
  4. Type the number
  5. You may have to press "=" depending on the model of the calculator.

Not all calculators have a "logy x" key, so see "change of base" in properties of logarithms below.

Working out the log of a number to the base 10 using an Android calculator app.

Working out the log of a number to the base 10 using an Android calculator app.

Working out the log of a number to the base 10 using an Android calculator app. Some calculators require you to press "=" after you enter the number.

Working out the log of a number to the base 10 using an Android calculator app. Some calculators require you to press "=" after you enter the number.

Exercise B: Calculate Values of Logs

Calculate the value of these logs:

  1. log2256
  2. log101000,000
  3. log381
  4. log1.5 3.375
  5. 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 ex is itself. So d/dx(ex) = ex

The log of a number x to the base e is normally written as ln x or logex

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 x0 = 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.
Graph of the log x to various bases.

Graph of the log x to various bases.

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.

    logc (AB) = logc A + logc 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.

    logc (A/B) = logc A - logc B

  • The power rule:

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

    logc (Ab) = blogcA

  • Change of base:

    logc A = logb A / logbc

    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 log2256 ?

    log2256 = log10256 / log102 = 8

Exercise C: Using Rules of Logs to Simplify Expressions

Simplify the following:

  1. log10 35x
  2. log10 5/x
  3. log10x5
  4. log10 10x3
  5. log2 8x4
  6. log3 27(x2/y4)
  7. log5 (1000) in terms of the base 10, rounded to two decimal places

What are Logarithms Used For?

  • Speeding up the multiplication and division of numbers
  • Representing numbers with a large dynamic range
  • Compressing scales on graphs
  • Simplifying functions to work out derivatives

The History of Logarithms

In the old days before the invention of mechanical, electromechanical and electronic calculators, multiplication and division of numbers with decimal paces was a tedious process. Logarithms were actually invented in the 17th century by the Scottish mathematician John Napier as a technique to speed up these calculations.

Multiplying numbers with logarithms

Imagine we have two numbers a and b.
We want to find the result of multiplying the two numbers, i.e. to find ab.
Take the log of ab and using the addition rule of logarithms:

log ab = log a + log b

Take the antilog of both sides

antilog(log ab) = antilog (log a + log b)

The antilog and log cancel, giving

ab = antilog (log a + log b)

So to multiply two numbers, you simply find the log of each number, add the results together and then find the antilog of the sum of the logs of the numbers. If a lot of numbers need to be multiplied together, the operation of multiplication is replaced by the much faster operation of addition.
Traditionally before calculators were invented, people used printed tables like the one in the image below and looked up the log of each of the multiplicands (the numbers being multiplied) in the table. After adding all the log values together, the antilog of the sum was looked up in an antilog table.
Now log tables don't allow you to check the log of any number. Instead they usually only have log values for numbers with a certain number of places of decimals, depending on accuracy required. For many applications, three places of decimals is adequate, so they will typically tabulate logs for values from 1.000 to 9.999 when using base 10.
So what happens if you need to multiply larger numbers?
Again imagine we have two numbers a and b, lets say both are greater than 10. We express them in scientific notation as a number less than 10 with a multiplier that is a power of 10 (e.g. 2348 = 2.348 x 103). Then when we take the log of that number, the coefficient (2.348 in this example) gives the decimal part of the log.

So log 2348 = log (2.348 x 103) = log 2.348 + log 103

= log 2.348 + 3

The number "3" is called the characteristic.
To find the log of 2.348, which is the decimal part of the log of 2348, known as the mantissa, we use a table like the one below. This gives logs for numbers from 1.000 to 9.999 (corresponding to log values from 0.000 to 1.000 rounded)

Log table.

Log table.

Log table

Log table

First we look for the number 2.3 in the vertical column at the left edge of the table. Then we follow across the row until we find the intersection point with the "4" column. This gives us a number 3692. Next we need to find the difference which needs to be added for the last decimal place "8" in 2.348. Looking across the row again to the intersection point with the "8" column on the far right of the table, we find the value is 15. This is added to 3692 giving 3707. The value 3707 is 0.3707 since the decimal point is omitted in the table for clarity. So the log of 2.348 = 0.3707
Adding the characteristic and mantissa together gives us:

log 2348 = 0.3707 + 3 = 3.3707

Example: What is 12,600 x 18.539 x 0.046?

Convert to scientific notification:

12,600 = 1.26 x 104
18.539 = 1.854 x 101 (rounded to 3 decimal places so result can be found in table)
0.046 = 4.6 x 10-2

Using log tables:

Log 1.26 x 104 = 4.1004

Log 1.854 x 101 = 1.2681

For numbers less than 1, the characteristic and mantissa are still added together to give the final result, but the sign of the characteristic must be taken into account.

So log 0.046 = log (4.6 x 10-2) = log 4.6 + log 10-2

The characteristic is log 10-2 = -2

From the table, the mantissa is log 4.6 = 0.6628

Adding the two together gives:

log 0.046 = log 4.6 + log 10-2 = 0.6628 + (-2) + = -1.3372

Adding the three logs together:

4.1004 + 1.2681 + (-1.3372) = 4.0313

Now we need to find the antilog of this value:

The antilog operation (for base 10) is effectively working out the value of 10 to the power of the value you want to find the antilog of. So we need to find the value of 104.0313
Now knowing the rules of exponentiation, 104.0313 = 104 x 100.0313
The antilog table gives us the antilog of 0.03013 (I.e. 100.03013 ) as 1.075.
So 104.0313 = 104 x 100.0313 = 10,000 x 1.075 = 1075

So finally

12,600 x 18.539 x 0.046 = 1075 approximately

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 = 20log10 ( p / p0)

where p is the pressure and po 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 = 20log10 ( p / p0)

= 20log10 ( 20 x 10-5 / 20 x 10-5)

= 20log10 (1) = 20 x 0 = 0dB

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

Then SPL = 20log10 ( p / p0)

= 20log10 ( 20 x 10-4 / 20 x 10-5)

= 20log10 (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 = 20log10 ( p / p0)

= 20log10 ( 20 x 10-3 / 20 x 10-5)

= 20log10 (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 log10 ( A / A0 ) where A is the amplitude and A0 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
Trip current for an MCB protective device. (These are used to prevent cable overload and overheating when excess current flows). The current scale and time scale are logarithmic.

Trip current for an MCB protective device. (These are used to prevent cable overload and overheating when excess current flows). The current scale and time scale are logarithmic.

Frequency response of a low pass filter, a device that only allows low frequencies through below a cut-off frequency (e.g. audio in a sound system). The frequency scale on the x axis and gain scale on the y axis are logarithmic.

Frequency response of a low pass filter, a device that only allows low frequencies through below a cut-off frequency (e.g. audio in a sound system). The frequency scale on the x axis and gain scale on the y axis are logarithmic.

Answers to Exercises

Exercise A

  1. y(a + b +c)
  2. p(a + b -x - y)
  3. p(a +b/q
  4. (ab)18
  5. a23b48

Exercise B

  1. 8
  2. 6
  3. 4
  4. 3
  5. 3

Exercise C

  1. log10 35 + log10 x
  2. log10 5 - log10x
  3. 5log10x
  4. 1 + 3log10x
  5. 3 + 4log2x
  6. 3 + 2log3x - 4log3y
  7. log10 1000 / log10 5 = 4.29 approx

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

© 2019 Eugene Brennan