4.5Logarithmic Properties
Apply the product, quotient, and power rules to expand and condense logarithmic expressions. Use the change-of-base formula to evaluate any logarithm with a calculator.
Logarithm properties are the algebraic backbone of solving exponential equations, simplifying calculus expressions, and working with information theory. Every time you solve for an unknown exponent, you are using these rules.
Essential Question
How do the product, quotient, and power rules for logarithms mirror the exponent rules — and why does understanding this connection make expanding and condensing logarithmic expressions feel automatic?
Lesson Overview
Logarithm properties are the algebraic tools that let us rewrite complex log expressions into simpler forms — and vice versa. They arise directly from exponent rules: because log_b(x) is an exponent, every exponent rule has a matching log rule. Mastering these three properties (product, quotient, power) plus the change-of-base formula unlocks the ability to solve exponential and logarithmic equations of any base.
All four logarithm properties — memorize these before moving on
Why These Rules Work
- Product: b^m · b^n = b^(m+n) → log of product = sum
- Quotient: b^m / b^n = b^(m−n) → log of quotient = difference
- Power: (b^m)^n = b^(mn) → exponent pulls out front
Restrictions
- All properties require same base on both sides
- Arguments M and N must be positive
- Base b: b > 0, b ≠ 1
- Power rule: n can be any real number
- Change of base: new base must also be valid
Expanding breaks one log into many; condensing reverses the process
Why the power rule works: repeated addition of the same log
Worked Examples
Expand completely: log₂(8x³/y)
Apply quotient rule: log₂(8x³/y) = log₂(8x³) − log₂(y)
Apply product rule: log₂(8x³) = log₂(8) + log₂(x³)
Apply power rule: log₂(x³) = 3·log₂(x)
Evaluate: log₂(8) = 3 (since 2³ = 8)
Combine: 3 + 3·log₂(x) − log₂(y)
Expand completely: ln(√x · y² / z⁵)
√x = x^(1/2), so rewrite: ln(x^(1/2) · y² / z⁵)
Quotient rule: ln(x^(1/2) · y²) − ln(z⁵)
Product rule: ln(x^(1/2)) + ln(y²) − ln(z⁵)
Power rule: (1/2)·ln(x) + 2·ln(y) − 5·ln(z)
Condense into a single logarithm: 3·log(x) + log(y) − 2·log(z)
Apply power rule in reverse: 3·log(x) = log(x³); 2·log(z) = log(z²)
Expression becomes: log(x³) + log(y) − log(z²)
Apply product rule: log(x³) + log(y) = log(x³y)
Apply quotient rule: log(x³y) − log(z²) = log(x³y/z²)
Condense: (1/2)·log₃(x) − 3·log₃(y) + log₃(z)
Power rule: (1/2)·log₃(x) = log₃(x^(1/2)) = log₃(√x)
Power rule: 3·log₃(y) = log₃(y³)
Expression: log₃(√x) − log₃(y³) + log₃(z)
Quotient rule: log₃(√x) − log₃(y³) = log₃(√x / y³)
Product rule: log₃(√x / y³) + log₃(z) = log₃(z√x / y³)
Use change of base to evaluate log₇(50) to four decimal places.
Change of base formula: log₇(50) = log(50) / log(7)
log(50) = log(5 · 10) = log(5) + log(10) = 0.6990 + 1 = 1.6990
log(7) ≈ 0.8451
log₇(50) ≈ 1.6990 / 0.8451 ≈ 2.0105
Verify: 7^2.0105 ≈ 50 ✓
Guided Practice
Expand completely: log₃(27x²/y)
Hint: Apply quotient rule first, then product rule, then power rule. Evaluate log₃(27) directly.
Expand completely: ln(x⁴ · √y / z³)
Hint: Rewrite √y = y^(1/2). Then apply quotient → product → power rules in order.
Condense into a single logarithm: 2·log₅(x) − log₅(y) + 3·log₅(z)
Hint: Apply power rule first to each term, then product/quotient rules to combine.
Condense: (1/3)·ln(x) + 2·ln(y) − ln(z)
Hint: Power rule: (1/3)·ln(x) = ln(x^(1/3)) = ln(∛x). Then combine with product and quotient rules.
Use change of base to evaluate log₄(30) to four decimal places.
Hint: log₄(30) = log(30)/log(4). Compute each with a calculator, then divide.
Key Vocabulary
Product Rule
log_b(MN) = log_b(M) + log_b(N). The log of a product equals the sum of the logs. Requires same base and positive arguments.
Quotient Rule
log_b(M/N) = log_b(M) − log_b(N). The log of a quotient equals the difference of the logs.
Power Rule
log_b(Mⁿ) = n·log_b(M). An exponent inside a log can be moved to a coefficient in front of the log.
Change of Base Formula
log_b(x) = log(x)/log(b) = ln(x)/ln(b). Converts any logarithm to base 10 or base e for calculator evaluation.
Expanding
Rewriting a single logarithm of a product, quotient, or power as a sum/difference of simpler logarithms.
Condensing
Rewriting a sum or difference of logarithms as a single logarithm. The reverse of expanding.
Coefficient
In the power rule, the exponent n becomes a coefficient (multiplier) in front of the log after the rule is applied.
Like Bases
All logarithm properties require that every log in the expression shares the same base before the rules can be applied.
Check Your Understanding
Interactive Practice — 5 Questions
Expand: log₂(x⁵y)
Condense: log(x) + log(y) − log(z)
Which expression equals 3·log₅(x) − (1/2)·log₅(y)?
Evaluate log₆(36) using the power rule.
Use change of base to evaluate log₃(15). Which setup is correct?
Independent Practice
Independent Practice
Expand completely: log₄(16x³/y²)
Expand completely: ln(x²√y / z⁴)
Condense into one logarithm: 4·log(x) + 2·log(y) − log(z)
Condense: (1/2)·log₂(x) − 3·log₂(y) + 2·log₂(z)
Use change of base to evaluate log₅(80) to four decimal places.
These are NOT valid log properties — a very common source of errors
Change of base gives the same answer whether you use base 10 or base e
Common Mistakes
Writing log(M + N) = log(M) + log(N) — splitting a sum inside a log.
The product rule applies to log(M · N), not log(M + N). There is no rule for the log of a sum.
Forgetting to apply the power rule before condensing: writing 3·log(x) + log(y) = log(3x + y).
First use power rule: 3·log(x) = log(x³). Then product rule: log(x³) + log(y) = log(x³y).
Applying change of base upside-down: log₃(15) = log(3)/log(15).
log_b(x) = log(x)/log(b). The argument goes on top, the base goes on the bottom.
Mixing bases: log₂(x) + log₃(y) = log₂(xy).
Product and quotient rules require the same base. You cannot combine logs with different bases directly.
Math Tips
Every log property mirrors an exponent rule. If you know b^m · b^n = b^(m+n), you already know the product rule.
Expand order: quotient rule first → product rule → power rule. Condense in reverse: power rule first → product/quotient rule last.
Change of base is your calculator bridge. Always put the argument on top: log_b(x) = log(x)/log(b).
There is NO rule for log(M + N) or log(M − N). If you see a sum inside a log, you cannot split it.
To verify a condensed answer, expand it back out — if you get the original expression, you are correct.