AP® Precalculus review sheet from Aim for Five (aimforfive.com/precalc/units/2/2-12)
Unit 2 · Topic 2.12
2.12 Logarithmic Function Manipulation
Log properties let you split one log into several or combine several into one. They come straight from the exponent rules. Each one also has a graph meaning, and the change-of-base rule shows that every log function is a stretch of every other.
Key terms
- product property
- quotient property
- power property
- change of base
The properties
For b > 0, b ≠ 1, and positive x and y:
| Property | Rule | Example |
|---|---|---|
| Product | log_b(xy) = log_b x + log_b y | log 50 = log 5 + log 10 |
| Quotient | log_b(x/y) = log_b x − log_b y | ln(e/2) = 1 − ln 2 |
| Power | log_b(xⁿ) = n · log_b x | log₂(x³) = 3 log₂ x |
| Change of base | log_b x = log_a x / log_a b | log₂ 10 = ln 10 / ln 2 |
Where they come from
Logs are exponents, so the log rules are the exponent rules in disguise. Multiplying powers adds exponents (bᵐ · bⁿ = bᵐ⁺ⁿ), so the log of a product is the sum of the logs. Raising a power to a power multiplies exponents, so the log of xⁿ is n times the log of x.
Graph meanings
Product property: log_b(kx) = log_b k + log_b x. A horizontal dilation of a log graph is the same as a vertical shift. For example, log₂(8x) = 3 + log₂(x).
Power property: log_b(xᵏ) = k · log_b x. Raising the input to a power is a vertical dilation. For example, ln(x²) = 2 ln x for x > 0.
Change of base: log_a x = (1/log_b a) · log_b x. Every log function is a vertical dilation of every other one, so they all have the same basic shape.
Natural log and calculators
ln x is log base e, so all the properties hold for ln too: ln(xy) = ln x + ln y, and so on.
Calculators have log (base 10) and ln (base e) keys. For another base, use change of base: log₅ 40 = ln 40 / ln 5 ≈ 2.292. Many calculators also have a logBASE command.
What the properties don't say
There's no rule for the log of a sum: log(x + y) is not log x + log y. And (log x)² is not 2 log x; the power rule needs the exponent inside, on x.
A quotient of logs is not the log of a quotient: log_b(x)/log_b(y) is not log_b(x/y). By change of base, it equals log_y(x).
The rules also need positive inputs. ln(x²) = 2 ln x holds only for x > 0. For every x ≠ 0, the correct version is ln(x²) = 2 ln|x|.
Using known log values
The properties let you build new values from known ones. If log₂ 3 ≈ 1.585, then log₂ 12 = log₂(4 · 3) = log₂ 4 + log₂ 3 ≈ 2 + 1.585 = 3.585.
A common question type gives letters for logs, such as a = ln 2 and b = ln 3, and asks for another log in terms of them. Since 18 = 2 · 3², ln 18 = ln 2 + 2 ln 3 = a + 2b.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Expanding
Expand log₃(9x²/y) completely, for x > 0 and y > 0.
Show the solutionHide the solution
- Step 1: Quotient rule: log₃(9x²) − log₃(y).
- Step 2: Product rule: log₃(9) + log₃(x²) − log₃(y).
- Step 3: log₃(9) = 2, and the power rule gives log₃(x²) = 2 log₃(x).
Answer: 2 + 2 log₃(x) − log₃(y)
- Example 2
Condensing
Write 2 ln x − ln(x + 1) + 3 ln 2 as a single logarithm, for x > 0.
Show the solutionHide the solution
- Step 1: Power rule first, to move the coefficients inside: ln(x²) − ln(x + 1) + ln(2³) = ln(x²) − ln(x + 1) + ln 8.
- Step 2: Combine the added logs with the product rule: ln(8x²) − ln(x + 1).
- Step 3: Quotient rule: ln(8x²/(x + 1)).
Answer: ln(8x²/(x + 1))
- Example 3
Trap: a sum inside the log
Is log(x + 10) equal to log(x) + 1? Is log(10x)?
Show the solutionHide the solution
- Step 1: log(x) + 1 = log(x) + log(10) = log(10x) by the product rule. So log(10x) = log(x) + 1 is true for x > 0.
- Step 2: For log(x + 10), test x = 10: log(20) ≈ 1.301, but log(10) + 1 = 2. Not equal.
- Step 3: There's no property that splits the log of a sum.
Answer: log(10x) = log(x) + 1, but log(x + 10) ≠ log(x) + 1.
Common mistakes
- Splitting the log of a sum or difference: log(x + y) ≠ log x + log y.
- Applying the power rule to (log x)². Only log(xⁿ) = n log x.
- Condensing before moving coefficients inside. Use the power rule first, then combine.
- Writing log_b x / log_b y as log_b(x/y). The quotient rule is about the log of a quotient, not a quotient of logs.
On the exam
- The symbolic-manipulation free-response question often asks you to rewrite log expressions as a single log or in terms of given logs. Show each property as you use it.
- Multiple-choice questions may ask which graph transformation is equivalent to a log rewrite, like log₂(8x) = 3 + log₂ x.
Connected topics
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Check yourself
4 questions on 2.12 Logarithmic Function Manipulation. Pick an answer to see if you got it, and why.
For x > 0 and y > 0, which of the following is equivalent to 2 ln x − ln(3y) + ln 6?
For x > 0, which of the following is equivalent to log₂(8x)?
Which of the following is equal to log₄ 10?
For x > 0 and y > 0, which of the following is equivalent to log₃(9x²/√y)?
0 of 4 answered