Properties of Log – Definition With Examples
Updated on September 29, 2026
The properties of logarithms are rules that let you rewrite and simplify logarithmic expressions. The three main ones are the product rule, log_b(MN) = log_b(M) + log_b(N); the quotient rule, log_b(M/N) = log_b(M) − log_b(N); and the power rule, log_b(Mⁿ) = n·log_b(M).
Logarithms — “logs” for short — are the inverse of exponents, and they appear in algebra, science, and computing. Their properties turn hard multiplication and division problems into simpler addition and subtraction. This is high-school-level material that high school math tutors cover in depth.
What Are Logarithms?
A logarithm answers the question: “What power must I raise the base to, in order to get this number?” If bʸ = x, then log_b(x) = y. For example, log₁₀(1000) = 3, because 10³ = 1000. In this way, logs are simply the inverse of exponents.
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The Properties of Logarithms
Logarithms follow a set of properties that make them easier to work with. The table below summarizes the three main rules plus three useful identities (here b is the base, M and N are positive numbers, and n is any real number).
| Property | Rule | Example |
| Product rule | log_b(MN) = log_b(M) + log_b(N) | log₁₀(6) = log₁₀(2) + log₁₀(3) |
| Quotient rule | log_b(M/N) = log_b(M) − log_b(N) | log₁₀(4) = log₁₀(8) − log₁₀(2) |
| Power rule | log_b(Mⁿ) = n · log_b(M) | log₁₀(2³) = 3 · log₁₀(2) |
| Log of 1 | log_b(1) = 0 | log₁₀(1) = 0 |
| Log of the base | log_b(b) = 1 | log₁₀(10) = 1 |
| Change of base | log_b(x) = log_c(x) / log_c(b) | log₂(8) = log₁₀(8) / log₁₀(2) |
Product, Quotient, and Power Rules Explained
Each of the three main rules matches a familiar operation — multiplication, division, or raising to a power. Working through a small example of each makes them click:
- Product rule: the log of a product equals the sum of the logs. So log₁₀(2 × 3) = log₁₀(2) + log₁₀(3).
- Quotient rule: the log of a quotient equals the difference of the logs. So log₁₀(8 ÷ 2) = log₁₀(8) − log₁₀(2).
- Power rule: the log of a number raised to a power equals the power times the log. So log₁₀(2³) = 3 × log₁₀(2).
Importance and Applications of Logarithms
Logarithms are more than a classroom idea — they power real calculations. They measure earthquake strength on the Richter scale, describe acidity as pH in chemistry, and appear throughout physics and engineering. In computer science, logarithms describe how efficient an algorithm is, and in finance, they help with compound interest.
Simplifying Expressions Using Log Properties
You can combine the rules to simplify a complicated log expression. Consider log₁₀(5³ × 2 ÷ 4): apply the product and quotient rules to get log₁₀(5³) + log₁₀(2) − log₁₀(4), then the power rule to get 3·log₁₀(5) + log₁₀(2) − 2·log₁₀(2), which simplifies to 3·log₁₀(5) − log₁₀(2). Going the other way, 3·log₁₀(5) − log₁₀(2) can be rewritten as log₁₀(5³ ÷ 2) = log₁₀(125/2).
Solving Equations Using Log Properties
Log properties also help solve equations. To solve log₂(x) + log₂(3) = 5, use the product rule to combine the left side into log₂(3x) = 5. That means 3x = 2⁵ = 32, so x = 32/3.
Solved Examples on Properties of Logarithms
These worked examples apply the rules step by step.
- Example 1: Simplify log₁₀(2 × 5²). Using the product and power rules, this becomes log₁₀(2) + 2·log₁₀(5), which is also log₁₀(50).
- Example 2: Simplify log₂((8 × 4) ÷ 2). Inside the log, (8 × 4) ÷ 2 = 16, and log₂(16) = 4.
- Example 3: Solve log₂(x²) + log₂(4) = 3. Combine to log₂(4x²) = 3, so 4x² = 8, x² = 2, and x = √2.
Practice Problems on Properties of Logarithms
Try these problems, then check the answers below.
- Simplify log₁₀(2 × 5²).
- Simplify log₂((8 × 4) ÷ 2).
- Solve for x in the equation log₂(x²) + log₂(4) = 3.
Answers: 1) log₁₀(2) + 2·log₁₀(5), which equals log₁₀(50). 2) 4. 3) x = √2.
Properties of Logarithms: Practice Test
Try these questions, then check the marked answers.
Properties of Logarithms Worksheets
Put this into practice with related Brighterly worksheets:
- Logarithmic Worksheets
- Exponents Worksheets
- Practice Worksheet Properties of Exponents Answer Key
- Algebra 2 Worksheets
Frequently Asked Questions About Properties of Logarithms
What Exactly Is a Logarithm?
A logarithm is the inverse of exponentiation. It tells you the power a base must be raised to in order to produce a given number. If bʸ = x, then log_b(x) = y. For example, log₁₀(100) = 2, because 10 raised to the power of 2 equals 100.
What Are the Properties of Logarithms?
The three main properties are the product rule, log_b(MN) = log_b(M) + log_b(N); the quotient rule, log_b(M/N) = log_b(M) − log_b(N); and the power rule, log_b(Mⁿ) = n·log_b(M). Two handy identities are log_b(1) = 0 and log_b(b) = 1.
Why Are the Properties of Logarithms Useful?
They turn hard operations into easier ones: multiplication becomes addition, division becomes subtraction, and powers become multiplication. This makes complex logarithmic expressions much simpler to compute and is a key tool for solving exponential and logarithmic equations.
How Are Logarithms Used in Real Life?
Logarithms appear in many fields. They measure earthquakes on the Richter scale, describe acidity through pH in chemistry, and are used across physics and engineering. In computer science they describe algorithm efficiency, and in finance they help calculate compound interest.
What Is the Change of Base Formula?
The change of base formula lets you rewrite a logarithm using a different base: log_b(x) = log_c(x) / log_c(b). It is useful because most calculators only compute logs in base 10 or base e, so you can convert any base into one of those to find a value.
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