An Introduction to Logarithms, Bases and Exponents
Learning About Logarithms
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 and using logarithms to perform multiplication and division.
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 255 = 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.
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
50 = 1
270 = 1
10000 = 1
2-4 = 1/24 = 1/16
10-3 = 1/103 = 1/1000
52 x 53 = 5 (2 + 3) = 55 = 3125
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 yaybyc
- Simplify papb/pxpy
- Simplify papb/qxqy
- Simplify (ab4)3 x (ab2)3
- Simplify ((ab4)3 x (ab4)3)2 / a25
Answers at bottom of page.
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 = 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 the subscript 10 in log10 . Sometimes this is 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
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.
23 = 8 and log2 8 = 3
34 = 81 and log3 81 = 4
Exercise B: Calculate Values of Logs
- Calculate log2256
- Calculate log101000,000
- Calculate log381
- Calculate log1.5 3.375
- Calculate 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.
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. A calculator normally only has "log" and "ln" keys for log to the base 10 and natural log to the base e respectively.
What is log2256 ?
log2256 = log10256 / log102 = 8
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.
What are Logarithms Used For?
- Representing numbers with a large dynamic range
- Compressing scales on graphs
- Multiplying and dividing decimals
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. 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
(1) y(a + b +c)
(2) p(a + b -x - y)
(3) p(a +b)/q(x + y)
© 2019 Eugene Brennan