A Review of Logarithms

Introduction and Overview

Reminder: What is a logarithm?

Does this look familiar?

• 10^2 = 100 This means ten squared, or ten to the second power, is 10 times10, is 100
• 10^3 = 1,000 This means ten cubed, or ten to the third power, is 10 times 10 times 10, is 1,000
• 10^4 = 10,000 This means ten to the fourth power, 10 times 10 times 10 times 10, is 10,000

A logarithm is the inverse of an exponent:

• The base 10 logarithm of 100 is 2 because 10^2 = 100
• The base 10 logarithm of 1,000 is 3 because 10^3 is 1,000
• The base 10 logarithm of 10,000 is 4 because 10^4 is 10,000
• What is the base 5 logarithm of 25?
• The base 5 logarithm of 25 is 2 because 5^2 is 25

The logarithm of a number is the exponent to which a base, such as e or 10, must be raised, to obtain that number. It is the power to which a number must be raised to produce a given value. If that sounds like a mouthful, it is. The quick examples above are more effective in explaining what a logarithm is.

When working with logarithms it is important to know what the base is. The preceding examples used base 10 and base 5. It is common to see base 2, e, 10, and 16. This tutorial will use base e and 10 almost exclusively. Base 10 logarithms are usually introduced first to students when learning logarithms, typically followed by base e and then other bases.

Definitions

e is Euler’s constant. It is approximately equal to 2.718282. It is frequently seen in
calculations involving population growth, investment yield, and other applications of
mathematics that use logarithms and exponents. e is a transcendental number,
meaning it can not be represented precisely using algebra. It is approximately equal to
the limit as x approaches infinity of (1 + 1/x)^x, which is approximately 2.718282.

Four Properties of logarithms

Logarithms have four properties. They are important to know in order to solve
problems when an unknown variable is in either an exponent or a logarithm.

1. The base ten log of 10 is 1 because 10^1 is 10. Using standard notation this
means that log(10) = 1 because 10^1 = 10

2. The log of 1 is zero because 10, or any other positive number, to the power of 0
is 1. In standard notation this is log(1) = 0 because 10^0 = 1

3. The base 10 log of 10 to any power is that power. For example, the base 10 log of
10^23 is 23. In standard notation this is log(10^23) = 23

4. Ten to the power of the base 10 log of any number is that number. For example
10^(log(23)) = 23

Applying the four properties of logarithms to base e logarithms.

Base e logarithms are also referred to as the natural log and use notation ln instead of log.

1. The base e log of e is 1 because e^1 is e. In standard notation this would be ln(e)
Eulers Constant

e^23 is 23. In standard notation this would be ln(e^23) = 23

4. e to the power of the base e log of any number is that number. For example
e^(ln(23)) = 23

Laws of Logarithms

There are three laws of logarithms:

1. The log of a product is the sum of the logs: log(AB) = log(A) + log(B)
2. The log of a quotient is the difference of the logs: log(A/B) = log(A) – log(B)
3. The log of an exponent is the power times the log: log(A^c) = c*log(A)

Log change of base formula

The formula to change the base of a log is

log_b(X) = log_a(X)/log_a(b)
Use this formula to convert a log expression from one base to another.

Knowing these properties, laws, and conversion rule; and how to use them, will allow
solving just about any kind of logarithm problem.

Why Study and Use Logarithms?

The difference between a minor and major earthquake is a variation in intensity on
the order of 10^8 or 100,000,000. The difference between the most quiet and loud
sound that people can hear is a factor of 23,000,000. The half-life of radioactive
materials varies between several seconds and thousands of years. These are all
examples of wide-ranging numbers that are more easily represented using logarithms.
It is for these reasons that logarithms are studied.

Using Logarithms

The rest of this tutorial is about how to use logarithms.