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 10^{2 }= 10 x 10 = 100
Similarly 4^{3} = 4 x 4 x 4 = 64
and 2^{5} = 2 x 2 x 2 x 2 x 2 = 32
We can also raise numbers with decimal parts (nonintegers) 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 rational number with a decimal part or even an irrational number such as the square root of two.
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
Exponent of 1
5^{1} = 5
2000^{1} = 2000
3.25^{1} = 3.25
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.
NonInteger 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 = 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 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:
log_{10} 5
In the specific case of log to the base 10, the subscript 10 is often 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
So in general
if c = a^{b}
then
log c = b
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}) = blog_{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?
 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 10^{3}). 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 10^{3}) = log 2.348 + log 10^{3 }
= 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)
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 10^{4 }18.539 = 1.854 x 10^{1} (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 10^{4} = 4.1004
Log 1.854 x 10^{1} = 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 10^{4.0313}
Now knowing the rules of exponentiation, 10^{4.0313} = 10^{4} x 10^{0.0313}
The antilog table gives us the antilog of 0.03013 (I.e. 10^{0.03013 }) as 1.075.
So 10^{4.0313} = 10^{4} x 10^{0.0313} = 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 = 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 xaxis or yaxis 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
 (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
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