3.3 Exponential, Logarithmic, and Piecewise Models

Exponential, Logarithmic, and Piecewise Models

Harder ACT Math function questions often ask you to recognize the model before doing computation. Linear models add a constant amount. Exponential models multiply by a constant factor. Logarithmic questions reverse exponent questions. Piecewise models change rules depending on the input. These are all in the official Functions description, so they are fair game for students aiming above the middle score range.

The section is calculator-permitted, and digital testers may have an on-screen graphing calculator, but these models still reward structure. The calculator can evaluate a power or confirm a graph. It cannot decide whether a situation is repeated addition, repeated multiplication, or a rule split by intervals.

Exponential Models: Repeated Multiplication

Use an exponential model when a quantity changes by the same percent or same factor each period. Growth often looks like y = a(1 + r)^t, where a is the starting amount, r is the growth rate as a decimal, and t is the number of periods. Decay often looks like y = a(1 - r)^t.

If a value increases by 8% each year, the factor is 1.08. If it decreases by 8% each year, the factor is 0.92. Do not multiply by 8 or subtract 8 from the output unless the question says a fixed number is added or removed. Percent change compounds from the current amount, not always from the original amount.

Worked example: a club has 240 members and grows by 5% each month. After t months, the model is M(t) = 240(1.05)^t. After 3 months, M(3) = 240(1.05)^3, about 278. This is not 240 + 3(12), even though the first month adds 12 members, because the second month adds 5% of a larger number.

Logarithms: Exponent Questions in Reverse

A logarithm asks for an exponent. The statement log_b(a) = c means b^c = a. On ACT Math, many logarithm questions are simple rewrites, domain checks, or equation solves. If you can translate the log into an exponent statement, the problem often becomes a familiar equation.

Worked example: log_2(32) = 5 because 2^5 = 32. If log_3(x - 1) = 4, rewrite as 3^4 = x - 1, so 81 = x - 1 and x = 82. The input x - 1 is positive at x = 82, so the solution is allowed.

The domain rule is nonnegotiable: the input of a logarithm must be positive. For log(x + 6), the domain is x > -6. For log(5 - x), the domain is x < 5. A tempting value that makes the log input zero is not allowed because no base raised to a power equals 0.

Piecewise Models: Choose the Rule Before Calculating

A piecewise function is one function built from several rules. The input decides which rule applies. ACT questions often make the arithmetic easy but punish students who use the wrong interval. Always compare the input to the condition before substituting.

Example:

For this function, f(-3) uses x^2 + 1, so f(-3) = 10. The value f(2) uses 3x - 2, so f(2) = 4. The value f(5) uses the constant rule, so f(5) = 10. The endpoints matter: x = 0 and x = 4 are included in the middle interval because the condition uses <=.

ACT Strategy and Common Traps

Use a model triage chart before calculating:

1. If equal x-steps produce equal y-differences, test a linear model.
2. If equal x-steps produce equal y-ratios, test an exponential model.
3. If the question asks what exponent is needed, rewrite the logarithm.
4. If the rule has intervals, mark the input on the number line first.
5. If a graph has a jump, open dot, or closed dot, check whether the endpoint is included.

The common exponential trap is treating percent change as a one-time percent of the original. A 20% decrease followed by a 20% increase does not return to the original value: 100 becomes 80, then 80 becomes 96. The base changed.

The common logarithm trap is accepting an algebraic solution that violates the domain. Solve the equation, then check the log input. If log(x - 4) appears, x = 4 is impossible and any x less than 4 is impossible.

The common piecewise trap is evaluating every rule and then choosing the nicest answer. The condition, not the output, selects the rule. If f(3) is requested and the interval says x >= 3, use that rule even if the neighboring rule would give a simpler number.

Recognizing Models from Tables and Graphs

A table can reveal the model before you write an equation. Equal differences point to linear change; equal ratios point to exponential change. Values 6, 12, 24, 48 multiply by 2 each step, so an exponential model fits. Values 6, 12, 18, 24 add 6 each step, so a linear model fits.

Graphically, exponential growth usually has a horizontal asymptote and then rises faster, while logarithmic growth rises quickly and then slows. ACT questions rarely require advanced graph theory here, but they do expect you to notice whether growth is speeding up, slowing down, or split into separate rules.