Unit 2 · Moderate

Exponential and Logarithmic Functions

A short unit with a single organising idea: exponential functions change by equal ratios over equal input intervals, and logarithms are the inverse that recovers the exponent. Most of the unit is consequences of that sentence, including modeling, equation solving, and the semi-log plot.

What a strong answer looks like

A strong Unit 2 answer names the growth factor and says what it means in context, rather than reporting a number stripped of its situation.

Practice this unit Course overview

Topics in this unit

Constant Ratios and Model Choice

Know

Linear functions add a constant over equal input intervals; exponential functions multiply by a constant. Comparing successive differences against successive ratios is what identifies the family from data.

Apply

Given a table, compute both differences and ratios. Whichever is (nearly) constant names the model.

Watch out

Data that grows quickly is not automatically exponential. A quadratic grows quickly too, and only the differences distinguish them.

Study move

Build two tables, one linear and one exponential, and confirm which of differences or ratios is constant in each.

Growth and Decay Factors

Know

A percentage change per period becomes a multiplicative factor: growth of r adds to one, decay of r subtracts from one. The factor, not the percentage, is what appears in the model.

Apply

Convert the stated percentage into a factor, then write the model with the exponent measured in the same periods as the factor.

Watch out

Using the percentage itself as the base is the standard error, and it produces catastrophic decay rather than modest growth.

Study move

Write models for 8% annual growth and 15% annual decay, then state each factor aloud in words.

Rescaling the Exponent

Know

When the given change happens over several periods rather than one, the exponent must be divided by the length of that span so the factor applies at the right rate.

Apply

If a quantity multiplies by k over p years, model it with k raised to the power t divided by p.

Watch out

Multiplying the exponent by the span instead of dividing compresses the growth dramatically, and the resulting numbers rarely look wrong enough to catch.

Study move

Model a population that grows from 500 to 800 in three years, then evaluate at t = 3 to confirm it returns 800.

Logarithms as Inverses

Know

A logarithm answers the question of what exponent produces a given value, which is why it inverts an exponential exactly. Its domain is the positive reals, because an exponential never outputs zero or a negative.

Apply

To find an inverse, swap the roles of input and output and solve. To find a domain, set the logarithm argument greater than zero.

Watch out

Restricting the argument to be at least zero rather than strictly greater is a boundary error that costs the endpoint.

Study move

Find the domain of a logarithm of a linear expression, then of a logarithm of a quadratic, where the answer is an interval union.

Logarithm Properties

Know

Logarithms convert products to sums, quotients to differences, and powers to coefficients. These three rules are the whole toolkit and they run in both directions.

Apply

Expand to separate a complicated argument, or condense to a single logarithm before exponentiating to solve.

Watch out

There is no rule for the logarithm of a sum. Splitting one is the most common invalid step in the unit.

Study move

Expand a logarithm of a product with an exponent, then condense the result back and confirm you recover the original.

Solving Exponential and Logarithmic Equations

Know

An exponential equation is solved by taking logarithms of both sides; a logarithmic equation by condensing and then exponentiating. The second route can introduce extraneous solutions.

Apply

After solving a logarithmic equation, substitute every candidate back and reject any that makes an argument non-positive.

Watch out

Skipping the domain check leaves an extraneous root in the answer, and the exam includes distractors built from exactly that omission.

Study move

Solve an equation combining two logarithms that yields two algebraic roots, only one of which survives.

Semi-Log Plots

Know

Plotting the logarithm of the output against the input linearises an exponential relationship, so exponential data appears as a straight line. The slope of that line relates to the growth factor.

Apply

Use a semi-log plot to decide whether data is exponential when the ordinary plot is ambiguous.

Watch out

A straight semi-log plot indicates exponential behavior, not linear behavior. The axis transformation is easy to forget when reading the result.

Study move

Take exponential data, take logarithms of the outputs, and confirm the transformed points fall on a line.

Emphasized in this unit

Connections and techniques that receive extra attention in this AP unit.

• Distinguishing exponential from polynomial growth by ratios rather than by how fast the numbers get large
• Converting a stated percentage into the multiplicative factor a model actually uses
• Dividing the exponent when the given change spans several periods
• Checking every logarithmic solution against the domain before reporting it

Varies by course

Related topics some schools attach to this unit and others leave out. Covered on request rather than assumed.

Natural logarithm emphasis. Some courses work mainly in base e; others stay with base 10 until calculus.
Compound interest. Frequently used as the motivating application, but the continuous case is not required.
Logistic models. Occasionally introduced alongside exponentials, though the framework does not require them.

Mastery checklist

• Identify an exponential model from constant ratios over equal input intervals.
• Convert a percentage growth or decay rate into a factor.
• Write a model when the given change spans more than one period.
• State the domain of a logarithmic expression.
• Expand and condense logarithms using the product, quotient, and power rules.
• Solve an exponential equation with logarithms and a logarithmic equation by exponentiating.
• Explain why exponential data is linear on a semi-log plot.

Check yourself

• Why is a constant ratio, not a large increase, the signature of an exponential?
• What goes wrong if you use 0.15 rather than 0.85 as a decay factor?
• Why can solving a logarithmic equation produce a root that must be rejected?
• What does the straightness of a semi-log plot tell you, and what does it not?

Modeling drill

A population is 500 at time zero and 800 after three years. Write the exponential model, state the annual growth factor, and say in one sentence what that factor means to someone running the study.

Constant ratioGrowth factorDecay factorLogarithmExtraneous solutionSemi-log plotInverse function

Unit 1: Polynomial and Rational Functions

Unit 3: Trigonometric and Polar Functions