Properties of Logarithms

What You’ll Learn

In this lesson you’ll learn the three main properties of logarithms and how to use them to simplify, expand, and condense logarithmic expressions.

Logarithms have three fundamental properties that make them powerful tools for simplification:

1. Product Property - the log of a product is the sum of the logs:

\log_b(MN) = \log_b(M) + \log_b(N)

2. Quotient Property - the log of a quotient is the difference of the logs:

\log_b\!\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)

3. Power Property - the log of a power brings the exponent out front:

\log_b(M^k) = k \cdot \log_b(M)

These properties work in both directions:

All logs must have the same base for these properties to apply.

1. Expand: log₂(8x³)

Use the product property, then the power property:

\log_2(8x^3) = \log_2(8) + \log_2(x^3) = 3 + 3\log_2(x)

2. Expand: log₅(25/y)

Use the quotient property:

\log_5\!\left(\frac{25}{y}\right) = \log_5(25) - \log_5(y) = 2 - \log_5(y)

3. Condense: 2log₃(x) + log₃(5) − 3log₃(y)

Use the power property first, then product and quotient:

\begin{aligned} &= \log_3(x^2) + \log_3(5) - \log_3(y^3) \\[1em] &= \log_3\!\left(\frac{5x^2}{y^3}\right) \end{aligned}

4. Simplify: log₁₀(1000) + log₁₀(0.1)

Use the product property to combine, then evaluate:

\log_{10}(1000 \times 0.1) = \log_{10}(100) = 2

Properties of logarithms are used to:

Example: in acoustics, the decibel scale is logarithmic. Properties of logs help combine multiple sound sources.

For reference, the three properties:

\log_b(MN) = \log_b(M) + \log_b(N)

\log_b\!\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)

\log_b(M^k) = k \cdot \log_b(M)

Using the product property: $\log_b(MN) =$

Condense: $3\log_2(x) - \log_2(5)$

The power property says $\log_b(M^k) =$

Which is equivalent to $\log_3(27x)$?

Expand: $\log_5\left(\frac{x^2}{y}\right)$