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log a·b·c

Using Logarithm
Properties

Master the product, quotient, and power properties to break apart and reassemble logarithmic expressions with confidence.

Algebra 2

What Is a Logarithm?

A logarithm answers the question: what exponent do I need? If \( b^x = N \), then \( \log_b N = x \). The logarithm is the inverse of exponentiation.

Logarithm properties let us rewrite complicated expressions into simpler pieces: products become sums, quotients become differences, and powers become multipliers.

Why Do Properties Matter?

Simplify
Break complex expressions like \( \log(x^3 y^2) \) into manageable parts.
Solve
Isolate variables trapped inside logarithms to solve equations.
Apply
Model real-world phenomena: earthquakes (Richter), sound (decibels), pH scales.

The Exponent-Log Connection

Exponent Rules
\( b^m \cdot b^n = b^{m+n} \)
\( \frac{b^m}{b^n} = b^{m-n} \)
\( (b^m)^n = b^{mn} \)
 Multiply bases → add exponents Divide bases → subtract exponents Log Properties
\( \log(MN) = \log M + \log N \)
\( \log\!\left(\frac{M}{N}\right) = \log M - \log N \)
\( \log(M^n) = n \cdot \log M \)
 Product → sum of logs Quotient → difference of logs

Each exponent rule translates directly into a logarithm property. Understanding one side helps you remember the other.

The Product Property

In words: the log of a product equals the sum of the logs. This works because multiplying numbers corresponds to adding their exponents.

The Quotient Property

In words: the log of a quotient equals the difference of the logs. Division of arguments becomes subtraction of logarithms.

The Power Property

In words: the log of a power equals the exponent times the log of the base argument. The exponent "drops down" and becomes a multiplier.

Expanding vs. Condensing

Expanding Condensing
Goal Break one log into several Combine several logs into one
Direction Left → Right of properties Right → Left of properties
Example \(\log\!\left(\frac{x^2}{y}\right) \to 2\log x - \log y\) \(3\log a + \log b \to \log(a^3 b)\)
When used Simplifying, analyzing Solving equations, graphing

How to Apply the Properties

1
Identify the operation inside the log: is it a product, quotient, or power?
2
Apply the matching property: products → sums, quotients → differences, powers → multiply in front.
3
Repeat until every log contains a single variable or number with no exponents.
4
Evaluate any logs you can compute exactly (e.g., \( \log_2 8 = 3 \)).

Example 1: Expanding a Log

1
Quotient Property: \( \log_3(x^4 y) - \log_3(z^2) \)
2
Product Property on the first term: \( \log_3(x^4) + \log_3 y - \log_3(z^2) \)
3
Power Property on \( x^4 \) and \( z^2 \): \( 4\log_3 x + \log_3 y - 2\log_3 z \)

Example 2: Condensing Logs

1
Power Property (move coefficients up): \( \log_2 a^3 - \log_2 b^{1/2} + \log_2 c \)
2
Product Property (add the + terms): \( \log_2(a^3 c) - \log_2 \sqrt{b} \)
3
Quotient Property (subtract → divide): \( \log_2\!\left(\frac{a^3 c}{\sqrt{b}}\right) \)

Example 3: Evaluate with Properties

1
Product Property: \( \log_6 4 + \log_6 9 = \log_6(4 \cdot 9) \)
2
Multiply: \( \log_6(36) \)
3
Evaluate: \( 6^2 = 36 \), so \( \log_6 36 = 2 \)

Decision Flowchart

See a log expression Expanding or Condensing? Expanding Condensing 1. Quotient → subtract 2. Product → add 3. Power → multiply 1. Coefficients → powers 2. Add → product 3. Subtract → quotient

Useful Identities

\( \log_b 1 = 0 \)
Any base raised to the 0 power equals 1.
\( \log_b b = 1 \)
Any base raised to the 1st power equals itself.
\( b^{\log_b x} = x \)
Inverse property: the exponential undoes the log.

Common Mistakes

✕ Distributing the log
\( \log(x + y) \neq \log x + \log y \). The product rule applies to \( \log(x \cdot y) \), not \( \log(x + y) \).
✕ Subtracting logs incorrectly
\( \log x - \log y \neq \log(x - y) \). It equals \( \log\!\left(\frac{x}{y}\right) \). Subtraction becomes division, not subtraction inside.
✕ Forgetting coefficient → power
When condensing, \( 2\log x \) must become \( \log x^2 \) before combining with other logs. Skipping this step gives wrong answers.

Summary of Properties

Property Rule Direction
Product \( \log_b(MN) = \log_b M + \log_b N \) Multiply ↔ Add
Quotient \( \log_b\!\left(\frac{M}{N}\right) = \log_b M - \log_b N \) Divide ↔ Subtract
Power \( \log_b(M^p) = p \cdot \log_b M \) Exponent ↔ Multiply
Identity \( \log_b 1 = 0 \), \( \log_b b = 1 \) Simplification

Question 1 of 5

Which expression is equivalent to \( \log_5(25x^3) \)?

Question 2 of 5

Condense: \( \log_3 12 - \log_3 4 \)

Question 3 of 5

Which is NOT a valid application of logarithm properties?

Question 4 of 5

Expand: \( \log_2\!\left(\frac{a^5}{b^3 c}\right) \)

Question 5 of 5

Condense: \( 2\ln x + \frac{1}{3}\ln y - 4\ln z \)