Logarithm Properties & Rules
All three core log rules (product, quotient, power), the change of base formula, cancellation laws, special values, expanding and condensing expressions, and six worked equation-solving examples. Complete precalculus coverage.
Quick Reference — All Log Properties
Product Rule
log₂(AB) = log₂ A + log₂ B
Multiplication inside → addition outside
Quotient Rule
log₂(A/B) = log₂ A − log₂ B
Division inside → subtraction outside
Power Rule
log₂(Aⁿ) = n · log₂ A
Exponent inside → multiplier outside
Change of Base
log₂(x) = ln(x) / ln(b)
Convert any base using ln or log₁₀
Log of Its Own Base
log₂(b) = 1
Because b¹ = b for any base
Log of 1
log₂(1) = 0
Because b⁰ = 1 for any base
Cancellation Law 1
log₂(bˣ) = x
Log and exponential cancel completely
Cancellation Law 2
b^(log₂ x) = x
Exponential and log cancel completely
1. Definition of a Logarithm
A logarithm is an exponent. The expression “log base b of x equals y” means exactly the same thing as “b to the y equals x.” The logarithm asks: to what power must I raise the base b to get x?
logb(x) = y ⇔ by = x
Logarithmic form and exponential form are two ways of saying the same thing.
Common Log (Base 10)
log(x)
No base written means base 10. Used on most calculators. Examples: log(100) = 2, log(1000) = 3, log(1) = 0.
Natural Log (Base e)
ln(x)
Base is Euler's number e ≈ 2.718. Essential for calculus. Examples: ln(e) = 1, ln(1) = 0, ln(e³) = 3.
Any Base b
logb(x)
Requires b > 0, b ≠ 1. Most common in precalculus besides base 10 and e. Example: log₂(8) = 3 because 2³ = 8.
Special Values Every Log Has
logb(1) = 0
The log of 1 is always 0 for any base, because b to the zero equals 1. In plain English: any number raised to the zero power gives 1, so the exponent (the log) is 0.
log(1) = 0 · ln(1) = 0 · log₇(1) = 0
logb(b) = 1
The log of the base itself is always 1, because b to the first power equals b. In plain English: you only need to raise b once to get b, so the exponent (the log) is 1.
log(10) = 1 · ln(e) = 1 · log₇(7) = 1
Converting between forms — practice examples
log₂(32) = 5 ⇔ 2⁵ = 32
log(10,000) = 4 ⇔ 10⁴ = 10,000
ln(e⁶) = 6 ⇔ e⁶ = e⁶
log₅(1/25) = −2 ⇔ 5⁻² = 1/25
log₃(81) = 4 ⇔ 3⁴ = 81
ln(1) = 0 ⇔ e⁰ = 1
2. The Three Core Logarithm Rules
These three rules are the workhorses of every log problem. They hold for any valid base b and any positive arguments A and B. Memorize the plain-English version of each rule — it's easier to recall under exam pressure.
logb(A · B) = logb A + logb B
When you have a product inside a logarithm, you can split it into a sum of two logs. This works in both directions: you can expand a log of a product, or you can condense a sum of logs back into a single log.
Why it works: this mirrors the exponential rule bⁿ times bᵐ equals b to the (m plus n). Since logs are exponents, adding exponents corresponds to multiplying the bases.
Expanding example:
log₂(8 · 4) = log₂(8) + log₂(4)
= 3 + 2 = 5
Check: log₂(32) = 5 ✓
Condensing example:
log(x) + log(x + 2)
= log(x(x + 2))
= log(x² + 2x)
logb(A / B) = logb A − logb B
When you have a fraction inside a logarithm, you can split it into the difference of two logs. This mirrors the exponential rule bⁿ divided by bᵐ equals b to the (m minus n).
Common mistake to avoid: log(A minus B) does NOT equal log(A) minus log(B). The quotient rule applies only to division inside the argument, not subtraction.
Expanding example:
log(1000/10) = log(1000) − log(10)
= 3 − 1 = 2
Check: log(100) = 2 ✓
Condensing example:
ln(x²) − ln(x + 1)
= ln(x² / (x + 1))
logb(An) = n · logb A
When the argument of a log is raised to a power, you can bring the exponent out front as a multiplier. This is the most important rule for solving exponential equations, because it lets you move the variable out of the exponent position.
The power rule works for any exponent n — positive, negative, or fractional. For example, a square root is an exponent of one-half, so log of the square root of A equals one-half times log A.
Expanding example:
log(10⁵) = 5 · log(10) = 5 · 1 = 5
ln(e²) = 2 · ln(e) = 2 · 1 = 2
log₂(x⁶) = 6 · log₂(x)
Fractional exponents:
log(√x) = log(x^(1/2)) = (1/2) · log(x)
ln(x^(1/3)) = (1/3) · ln(x)
Domain requirement for all three rules
All three rules require that A and B are strictly positive. You cannot apply log rules to negative arguments or zero — those are outside the domain of any logarithmic function. When expanding or condensing, always confirm the domain of the resulting expression matches.
3. Change of Base Formula
Most calculators only have buttons for base-10 log and natural log (ln). The change of base formula lets you evaluate or rewrite any logarithm using those two bases.
logb(x) = log(x) / log(b) = ln(x) / ln(b)
Both forms give the same answer. Use whichever base your calculator supports.
Worked Examples — Change of Base
Evaluate log₂(10)
log₂(10) = ln(10) / ln(2)
≈ 2.302585 / 0.693147
≈ 3.3219
Check: 2 to the 3.3219 ≈ 10 ✓
Evaluate log₅(200)
log₅(200) = log(200) / log(5)
≈ 2.30103 / 0.69897
≈ 3.292
Check: 5 to the 3.292 ≈ 200 ✓
Rewrite log₃(x) using ln
log₃(x) = ln(x) / ln(3)
Useful when differentiating in calculus
Simplify log₄(16)
log₄(16) = log(16) / log(4)
= log(4²) / log(4)
= 2 · log(4) / log(4) = 2
Or: 4² = 16 directly, so log₄(16) = 2
4. Cancellation Laws (Inverse Properties)
Because the exponential function y equals b to the x and the logarithm y equals log base b of x are inverse functions, applying one and then the other gives back the original input. These cancellation laws are among the most frequently tested identities in precalculus.
Cancellation Law 1
logb(bx) = x
Log base b of b to the x equals x, for all real numbers x. The logarithm and the exponential cancel each other out completely.
log₂(2⁵) = 5
log(10⁻²) = −2
ln(e̯) = k
log₅(5⁵⁄²) = 9/2
Cancellation Law 2
blogb(x) = x
b to the power log base b of x equals x, for all positive x. The exponential and the logarithm cancel each other out completely.
2^(log₂ 7) = 7
10^(log 5.3) = 5.3
e^(ln x) = x
3^(log₃ 100) = 100
Natural Log Special Cases
Because natural log and the base-e exponential are the most common pair in calculus, their cancellation laws are worth memorizing separately:
ln(eE3;) = x (for all real x)
e^(ln x) = x (for x > 0)
ln(e) = 1
ln(1) = 0
ln(e²) = 2
ln(e⁻¹) = −1
5. Expanding and Condensing Log Expressions
Expanding means breaking a single log of a complex expression into a sum or difference of simpler logs. Condensing means combining multiple logs into a single log. Both use the same three rules — you just apply them in opposite directions.
Expanding Examples
Expand: log₂(x⁵ times y² divided by z)
log₂(x⁵ · y² / z)
= log₂(x⁵) + log₂(y²) − log₂(z) product and quotient rules
= 5 · log₂(x) + 2 · log₂(y) − log₂(z) power rule
Expand: ln(square root of (x divided by (y times z²)))
ln(√(x / (y · z²)))
= ln((x / (y · z²))^(1/2))
= (1/2) · ln(x / (y · z²)) power rule
= (1/2) · [ln(x) − ln(y · z²)] quotient rule
= (1/2) · [ln(x) − ln(y) − 2 · ln(z)] product and power rules
= (1/2)ln(x) − (1/2)ln(y) − ln(z)
Expand: log((x + 1)³ divided by (x times (x − 2)²))
log((x + 1)³ / (x(x − 2)²))
= log((x + 1)³) − log(x(x − 2)²) quotient rule
= log((x + 1)³) − [log(x) + log((x − 2)²)] product rule
= 3 · log(x + 1) − log(x) − 2 · log(x − 2) power rule
Condensing Examples
Condense: 3 · log(x) + 2 · log(y) − log(z)
3 · log(x) + 2 · log(y) − log(z)
= log(x³) + log(y²) − log(z) power rule (move coefficients in)
= log(x³ · y²) − log(z) product rule
= log(x³y² / z) quotient rule
Condense: (1/2) · ln(x) − 2 · ln(y) + ln(3)
(1/2) · ln(x) − 2 · ln(y) + ln(3)
= ln(x^(1/2)) − ln(y²) + ln(3) power rule
= ln(3 · x^(1/2)) − ln(y²) product rule
= ln(3√x / y²) quotient rule
Condense: log₂(x + 3) + log₂(x − 3)
log₂(x + 3) + log₂(x − 3)
= log₂((x + 3)(x − 3))
= log₂(x² − 9) difference of squares
Domain: both arguments must be positive, so x > 3 (not just x > −3).
6. Solving Equations — Six Worked Examples
Log equations fall into two broad categories: (A) solving for a variable inside a log by converting to exponential form, and (B) solving exponential equations where the variable is in the exponent by taking a log of both sides. Both types require checking domain.
Always check for extraneous solutions
Log equations can produce values that satisfy the algebra but make a log argument zero or negative. These are extraneous and must be discarded. After solving, substitute back into the original equation and verify every log argument is strictly positive.
Single log — convert to exponential form
Solve: log₃(x) = 4
log₃(x) = 4
Convert to exponential form: 3⁴ = x
x = 81
Check: log₃(81) = 4 because 3⁴ = 81 ✓
Isolate the log first, then convert
Solve: 2 · log(x) − 3 = 5
2 · log(x) − 3 = 5
Add 3: 2 · log(x) = 8
Divide by 2: log(x) = 4
Exponential form: 10⁴ = x
x = 10,000
Check: 2 · log(10000) − 3 = 2(4) − 3 = 5 ✓
Multiple logs — combine first, then convert
Solve: log(x) + log(x + 3) = 1
log(x) + log(x + 3) = 1
Product rule: log(x(x + 3)) = 1
Exponential form: x(x + 3) = 10
Expand: x² + 3x − 10 = 0
Factor: (x + 5)(x − 2) = 0
So x = −5 or x = 2
Check x = −5: log(−5) undefined — discard
x = 2 Check: log(2) + log(5) = log(10) = 1 ✓
Log equals log — set arguments equal
Solve: log₂(3x + 4) = log₂(x + 10)
Same base on both sides → set arguments equal:
3x + 4 = x + 10
2x = 6
x = 3
Check: log₂(13) = log₂(13) ✓ (both args positive ✓)
Variable in the exponent — take log of both sides
Solve: 5E3; = 80
5E3; = 80
Take ln of both sides: ln(5E3;) = ln(80)
Power rule: x · ln(5) = ln(80)
Divide: x = ln(80) / ln(5)
x = ln(80) / ln(5) ≈ 4.3820 / 1.6094 ≈ 2.723
Or equivalently: x = log(80) / log(5) ≈ 2.723
Natural log equation with cancellation
Solve: e^(2x − 1) = 9
e^(2x − 1) = 9
Take ln of both sides: ln(e^(2x − 1)) = ln(9)
Cancellation law: 2x − 1 = ln(9)
Add 1: 2x = 1 + ln(9)
x = (1 + ln(9)) / 2 ≈ (1 + 2.197) / 2 ≈ 1.599
7. Natural Log Properties
The natural logarithm ln is simply the logarithm with base e (Euler's number, e ≈ 2.71828). Every property that applies to log base b applies to ln with b replaced by e. Because e is the natural base for calculus, ln appears far more often than other bases in advanced problems.
| Property | Formula | Example |
|---|---|---|
| ln of 1 | ln(1) = 0 | because e⁰ = 1 |
| ln of e | ln(e) = 1 | because e¹ = e |
| Product rule | ln(AB) = ln A + ln B | ln(6) = ln(2) + ln(3) |
| Quotient rule | ln(A/B) = ln A − ln B | ln(e²/e) = 2 − 1 = 1 |
| Power rule | ln(Aⁿ) = n · ln A | ln(e⁵) = 5 · ln(e) = 5 |
| Cancellation 1 | ln(eE3;) = x | ln(e̯) = k, all real k |
| Cancellation 2 | e^(ln x) = x | e^(ln 7) = 7, x > 0 |
| Change of base | log₂(x) = ln(x) / ln(b) | log₂(8) = ln(8)/ln(2) = 3 |
8. Graphing Logarithmic Functions
Understanding the graph of y equals log base b of x is essential for recognizing domain restrictions, identifying the vertical asymptote, and understanding the inverse relationship with exponential functions.
y = logb(x) with b > 1
- Domain: x > 0, written (0, ∞)
- Range: all real numbers (−∞, ∞)
- x-intercept: (1, 0) — because log₂(1) = 0
- No y-intercept — curve never crosses the y-axis
- Vertical asymptote: x = 0 (the y-axis)
- Behavior: increasing, but grows very slowly as x increases
- Key point: passes through (b, 1) because log₂(b) = 1
y = logb(x) with 0 < b < 1
- Domain: x > 0, written (0, ∞)
- Range: all real numbers (−∞, ∞)
- x-intercept: (1, 0) — same as b > 1 case
- Vertical asymptote: x = 0 (the y-axis)
- Behavior: decreasing — graph falls as x increases
- Example: y = log(1/2)(x) is a reflection of y = log₂(x) over the x-axis
Inverse Relationship with Exponential Functions
The functions y equals b to the x and y equals log base b of x are inverses of each other. On a graph, they are reflections of each other over the line y equals x. This means:
- Every feature swaps: domain ↔ range, x-intercept ↔ y-intercept, horizontal asymptote ↔ vertical asymptote
- If the point (a, b) is on y equals b to the x, then (b, a) is on y equals log base b of x
- Example: (3, 8) is on y equals 2 to the x, so (8, 3) is on y equals log₂(x)
y = bE3;: domain all reals, range (0,∞)
y = log₂(x): domain (0,∞), range all reals
y-intercept (0,1)
x-intercept (1,0)
horizontal asymptote y = 0
vertical asymptote x = 0
Shifted and Transformed Logs
When a log function is shifted or scaled, the asymptote and domain shift too. Always identify where the argument of the log equals zero — that is where the vertical asymptote goes, and you need x to be on the side that makes the argument positive.
y = log₂(x − 3)
Shift right 3 units. Asymptote at x = 3. Domain: x > 3.
y = log₂(x + 5)
Shift left 5 units. Asymptote at x = −5. Domain: x > −5.
y = −log₂(x)
Reflection over x-axis. Asymptote still x = 0. Domain still x > 0.
Frequently Asked Questions
What are the three main logarithm properties (rules)?
The three core log rules are: (1) Product Rule — the log of A times B equals log A plus log B. Multiplication inside the log becomes addition outside. (2) Quotient Rule — the log of A divided by B equals log A minus log B. Division inside becomes subtraction outside. (3) Power Rule — the log of A to the power n equals n times log A. An exponent inside the log comes out front as a multiplier. These three rules, together with the change of base formula, let you expand, condense, or rewrite any logarithmic expression.
What is the change of base formula and when do you use it?
The change of base formula converts a logarithm in any base to a ratio of logarithms in a convenient base: log base b of x equals log x divided by log b, which also equals ln x divided by ln b. You use it whenever your calculator only has base-10 or base-e buttons but the problem uses a different base. For example, log base 2 of 10 equals ln(10) divided by ln(2), which is approximately 3.322. Both forms give the same answer.
What are the cancellation laws for logarithms?
The two cancellation laws express the inverse relationship between exponential and logarithmic functions: (1) log base b of b to the x equals x, for all real x. The log and the exponential cancel completely. (2) b to the log base b of x equals x, for all positive x. These work because log base b and the base-b exponential are inverse functions. Special cases: ln of e to the x equals x, and e to the ln x equals x. These identities are essential for simplifying expressions and solving equations.
How do you solve a logarithmic equation?
The key step is converting between logarithmic and exponential form: log base b of x equals y means exactly the same thing as b to the y equals x. To solve a log equation: (1) Isolate the logarithm on one side. (2) Rewrite in exponential form and solve for the variable. (3) Check that your answer makes all log arguments positive — negative arguments and zero are outside the domain. For equations with multiple logs, first combine them using the product, quotient, or power rule to get a single log, then convert to exponential form.
What special values do logarithms always have?
Every logarithm base b (where b is positive and not equal to 1) has two guaranteed special values: (1) log base b of 1 equals 0 — because b to the zero equals 1 for any base. (2) log base b of b equals 1 — because b to the first equals b. These hold for common log (base 10), natural log (base e), and every other valid base. For natural log specifically: ln(1) equals 0, and ln(e) equals 1.
What is the domain of a logarithmic function and why does it matter?
The domain of y equals log base b of x is all positive real numbers: x must be strictly greater than zero. You cannot take the log of zero (undefined) or the log of a negative number (not a real number). This matters when solving equations because algebraic steps can produce extraneous solutions — values that satisfy the algebra but fall outside the domain. Always substitute your answers back into the original equation and discard any value that makes a log argument zero or negative.
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