Exponential & Logarithmic Functions

What this unit covers

The topics below follow the published Precalculus course framework for Unit 2. This unit is worth 27–40% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Exponential Growth & Decay12 min · 3 objectivesIdentify the initial value and base of an exponential function · Classify a function as growth or decay from its base · Build an exponential model from a doubling or halving description
• Properties of Logarithms13 min · 3 objectivesInterpret a logarithm as the exponent that produces a given number · Apply the product, quotient, and power rules for logarithms · Evaluate simple logarithms without a calculator
• Inverse Functions14 min · 3 objectivesExplain how exponential and logarithmic functions are inverses · Find the inverse of a basic exponential function · Use the fact that composing a function with its inverse returns the input
• Modeling with Exponentials & Logs13 min · 3 objectivesApply half-life reasoning to a decaying quantity · Choose an appropriate model for a real-world growth or decay scenario · Solve an exponential equation using logarithms

Formulas in Unit 2

Every term in Unit 2

All 30 terms we publish for Exponential & Logarithmic Functions, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Change of base formula

log_b(x) = log(x)/log(b), using any consistent base. Needed because calculators offer only base 10 and base e.

Semi-log plot

Plotting log(y) against x linearises exponential data. A straight line on a semi-log plot is evidence the underlying model is exponential.

Exponential function

f(x) = ab^x with a ≠ 0 and b > 0, b ≠ 1. Grows by a constant FACTOR per unit input, unlike a linear function's constant difference.

Recognizing exponential data

Equal input intervals produce a constant RATIO of outputs. Constant differences indicate linear; constant second differences indicate quadratic.

The number e

The base for which the function equals its own rate of change, about 2.71828. Arises as the limit of (1 + 1/n)ⁿ.

Exponential growth vs decay

b > 1 gives growth, 0 < b < 1 gives decay. Equivalently, a positive exponent coefficient grows and a negative one decays.

Horizontal asymptote of an exponential

y = 0 for the parent function; a vertical shift by k moves it to y = k. Exponentials approach but never reach it.

Logarithm definition

log_b(x) = y means b^y = x. A logarithm is an exponent — the single most useful sentence in this unit.

Domain of a logarithm

x > 0 strictly. The log of zero or a negative number is undefined over the reals, which is why solutions must be checked after solving.

Product and quotient rules for logs

log(MN) = log M + log N and log(M/N) = log M − log N. Valid only when M and N are both positive.

Power rule for logs

log(Mᵖ) = p·log M. The rule that turns an unknown exponent into a coefficient, which is how exponential equations get solved.

Solving exponential equations

Take the log of both sides and apply the power rule, or rewrite both sides with a common base and equate exponents.

Solving logarithmic equations

Condense to a single log, exponentiate, then CHECK every solution — squaring and condensing can introduce values outside the domain.

Extraneous solutions

Values satisfying the transformed equation but not the original. Common with logs (negative arguments) and with squaring both sides.

Inverse relationship of exp and log

b^(log_b x) = x and log_b(b^x) = x. Their graphs are reflections across y = x, so the exponential's horizontal asymptote becomes the log's vertical one.

Log-log plot

Plotting log(y) against log(x) linearises a power function y = axⁿ, and the slope of that line is n.

Compound interest

A = P(1 + r/n)^(nt) for n compoundings per year; A = Pe^(rt) compounded continuously.

Half-life and doubling time

The constant time for a quantity to halve or double, independent of starting amount — the defining property of exponential change.

Logistic model

Growth that is nearly exponential at first and levels off at a carrying capacity, giving an S-shaped curve with an inflection point at half the capacity.

Newton's law of cooling

Temperature difference from the surroundings decays exponentially, so the object cools fastest when the difference is largest.

Exponent rules

b^m·b^n = b^(m+n), b^m/b^n = b^(m−n), (b^m)^n = b^(mn), b^(−n) = 1/b^n, b^0 = 1.

Fractional exponents

b^(m/n) is the nth root of b^m. Converting radicals to exponents makes the exponent rules available.

Why the base cannot be negative

A negative base would make b^(1/2) undefined over the reals, so exponential functions require b > 0.

Transformations of exponentials

y = a·b^(x − h) + k: k moves the horizontal asymptote, h shifts horizontally, and a reflects when negative.

Natural log

ln x is log base e. Its derivative properties make it the base of choice in calculus, and it obeys all the ordinary log rules.

Common modeling error

Fitting an exponential to data that only looks curved. Check whether ratios of successive outputs are constant before choosing the model.

Residuals

Observed minus predicted. A good model leaves residuals scattered randomly about zero; a pattern in the residuals means the model is the wrong shape.

Interpreting model parameters in context

In y = ab^t, a is the initial amount and b − 1 is the fractional change per unit time. Both need units and a sentence.

Exponential vs power function

In an exponential the variable is the exponent; in a power function it is the base. 2^x and x² behave completely differently for large x.

Inverse of an exponential model

Solving y = ab^t for t gives t = log(y/a)/log(b) — the function that answers "when does it reach this value".

What examiners penalize here

• For growth described by a doubling/halving time T, use base 2 (or 1/2) with exponent t/T. For a percent rate r per period, use base (1 + r). Matching the base and the exponent to the description is the graded step.
• Use the power rule to solve exponential equations: log(bˣ) = x·log b lets you pull the variable out of the exponent. Recognizing when to expand versus condense is what the free-response rubric rewards.
• Use the composition identity to simplify: e^(ln x) = x and log(10ˣ) = x. Spotting a function next to its inverse lets you collapse a messy expression in one step.
• Pick the log base that simplifies the work: natural log (ln) pairs with base e, common log with base 10, but log₃ 20 = log 20 / log 3 works for any base via the change-of-base formula. State the exact answer before rounding.

Practice Precalculus

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 4 units of AP Precalculus

1. Unit 1 · Polynomial & Rational Functions
2. Unit 2 · Exponential & Logarithmic Functions
3. Unit 3 · Trigonometric & Polar Functions
4. Unit 4 · Functions Involving Parameters, Vectors & Matrices