- 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

- 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

more effective in explaining what a logarithm is.

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

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.

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

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

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_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.

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.

- Measure Sound Intensity
- Understanding the Richter Scale
- Estimate Radioactive Decay
- Forecasting Population Growth
- Using pH
- Computing Interest Rates