Laws of Logarithms

Logarithms help us deal with very large or very small numbers by turning multiplication and division into addition and subtraction. They are closely related to powers (indices), and just like powers, logarithms follow certain rules called laws of logarithms.

 

What is a Logarithm?

The logarithm of a number is the power to which you raise a base to get that number.

If $a^x = b$, then $\log_a b = x$

Example: Since $2^3 = 8$, then $\log_2 8 = 3$

 

1. Product Law

$$\log_a (MN) = \log_a M + \log_a N$$

When multiplying numbers inside a logarithm, you can split them into a sum of logs.

Example: $\log_{10}(100 \times 1000) = \log_{10}100 + \log_{10}1000 = 2 + 3 = 5$

 

2. Quotient Law

$$\log_a \left(\frac{M}{N}\right) = \log_a M - \log_a N$$

When dividing numbers inside a log, it becomes a subtraction of two logs.

Example: $\log_{10}\left(\frac{1000}{100}\right) = \log_{10}1000 - \log_{10}100 = 3 - 2 = 1$

 

3. Power Law

$$\log_a (M^k) = k \log_a M$$

A power inside the log can be brought to the front as a multiplier.

Example: $\log_{10}(100^2) = 2 \log_{10}(100) = 2 \times 2 = 4$

 

4. Change of Base Formula (for extension)

$$\log_b M = \frac{\log_a M}{\log_a b}$$

Used to convert logs from one base to another. This is helpful for calculator work or solving with different bases.

 

Worked Example 1

Worked Example 2

Practice Problem