AP PRECALCULUS — UNIT 2B · EXPONENTIAL & LOGARITHMIC FUNCTIONS
Unit 2B — Review
Notes — Concept Synthesis Across Topics 2.9 – 2.15
💡 Review Goals
By the end of this review you will be able to:
- Pull together every skill from 2.9 – 2.15 into a single mental framework
- Recognize which topic a question is testing from its surface features
- Avoid the most common errors that AP graders flag in this unit
- Set up and execute a clean solution to a multi-step modeling problem
1. The Unit at a Glance
Unit 2B builds the inverse companion to the exponentials of Unit 2A. The core insight: every fact about exponential functions has a mirror-image fact about logarithmic functions. Study them as a pair.
Topic | Big Idea |
|---|---|
2.9 | Logarithm = the exponent. log_b(y) = x ⇔ b^x = y. |
2.10 | log_b is the inverse of b^x; their graphs reflect across y = x. |
2.11 | Domain x > 0; vertical asymptote at 0; key point (1, 0); slow concave-down growth for b > 1. |
2.12 | Three properties (product, quotient, power) plus change-of-base. |
2.13A | Use one-to-one and convert log↔exp form to solve. |
2.13B | Take a logarithm of both sides when bases cannot be matched. |
2.14 | Logarithmic models: y = a + b log_c(x); used on dB, pH, Richter, and similar scales. |
2.15 | Semi-log plots linearize exponential data; slope = log(b), intercept = log(a). |
2. The Most Important Identities to Memorize
- log_b(y) = x ⇔ b^x = y
- log_b(b^x) = x and b^(log_b(x)) = x
- log_b(MN) = log_b(M) + log_b(N)
- log_b(M/N) = log_b(M) − log_b(N)
- log_b(M^k) = k · log_b(M)
- log_b(x) = log(x) / log(b) = ln(x) / ln(b) (change of base)
- log_b(1) = 0 and log_b(b) = 1
3. Decision Tree for Solving
Most equations in this unit can be classified into a few categories. Here is a quick decision tree:
- BOTH SIDES CAN BE WRITTEN WITH SAME BASE → equate exponents (2.13A)
- BOTH SIDES ARE A SINGLE LOG WITH SAME BASE → equate arguments (2.13A)
- ONE SIDE IS A LOG, OTHER SIDE IS A NUMBER → convert to exponential form (2.13A)
- MULTIPLE LOGS ON ONE SIDE → condense first using properties, then re-classify (2.13A/B)
- EXPONENTIAL WITH MISMATCHED BASES → take ln of both sides, use power rule (2.13B)
- QUADRATIC IN e^x OR b^x → substitute u = e^x, solve quadratic, back-substitute (2.13B)
4. Top Five AP-Style Pitfalls
⚠️ Mistakes to avoid
- Forgetting to check domain after condensing logs (always plug answers back into the ORIGINAL equation).
- Assuming log distributes over addition: log(M + N) ≠ log(M) + log(N).
- Confusing concave down with decreasing: log_b(x) for b > 1 is increasing AND concave down.
- Reversing inequality direction when not needed (or failing to reverse it when base < 1).
- Reading the y-axis on a semi-log plot as if it were linear — those tick marks represent powers of the base.
5. Modeling Workflow
A typical FRQ-style modeling problem in this unit follows this rhythm:
- Identify the family: linear (constant additive change), exponential (constant multiplicative change), or logarithmic (constant additive change per multiplicative change).
- Set up an equation in general form (e.g. y = a · b^x or y = a + b log(x)).
- Plug in given data points to solve for the parameters.
- Use the model to answer the question — predict, find a doubling/half-life time, or compare to another model.
- Interpret the parameters or answer in context (sentence form).
- Check that solutions are valid (positive populations, nonnegative time, etc.).
6. Worked Mini-Example
📘 Example — Multi-step problem
- A water sample has [H⁺] = 4.0 × 10⁻⁵. Find the pH.
- pH = −log(4.0 × 10⁻⁵) = −(log 4 + log 10⁻⁵) = −(0.602 − 5) ≈ 4.398
- Suppose a chemical change increases [H⁺] by a factor of 1000. The new pH is the old pH minus log(1000) = 3, so pH ≈ 1.398.
- Interpretation: a thousandfold increase in hydrogen-ion concentration corresponds to a 3-unit drop in pH — the hallmark of a logarithmic scale.
7. Final Reminders Before the Test
- Always state the domain conditions when you START — they prevent extraneous solutions later
- Show every property application by name on FRQs (‘by the product rule…’) — graders look for it
- On semi-log questions, check whether the slope is positive (growth) or negative (decay) before computing parameters
- Memorize that log(1) = 0 and ln(1) = 0 — they appear in nearly every problem
- Use units in your final answers when contextual