Exponential and Logarithmic Functions: every key term you need (+ practice quiz)
38 flashcard terms for AP Precalculus Unit 2, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 21-question quiz — free, no account needed.
A sequence with a constant difference d between consecutive terms: a_n = a_0 + dn (or a_k + d(n − k)); it is the discrete version of a linear function.
Geometric Sequence
A sequence with a constant ratio r between consecutive terms: g_n = g_0 · r^n; it is the discrete version of an exponential function.
Exponential Function
f(x) = a·b^x with a ≠ 0, b > 0, b ≠ 1; the output changes by a constant ratio b over each unit change in input.
Initial Value
In f(x) = a·b^x, a = f(0) is the output at x = 0 and the y-intercept of the graph.
Growth Factor
The base b of an exponential function; b > 1 means growth and 0 < b < 1 means decay. Growth rate r satisfies b = 1 + r.
Exponential Growth
When b > 1 in f(x) = a·b^x with a > 0, outputs increase and the graph is increasing and concave up, eventually outpacing any polynomial.
Exponential Decay
When 0 < b < 1, outputs shrink toward zero by a constant proportion each step; the graph is decreasing and concave up.
Horizontal Asymptote of an Exponential
The basic exponential a·b^x approaches y = 0 as x → −∞ (growth) or x → ∞ (decay); a vertical shift k moves the asymptote to y = k.
Equal Inputs, Proportional Outputs
The defining property of exponentials: over any interval of fixed length, the output is multiplied by the same factor, so f(x + c)/f(x) is constant.
Natural Base e
e ≈ 2.71828, the limit of (1 + 1/n)^n as n → ∞; f(x) = e^x is the natural exponential and models continuous growth.
Continuous Compounding
A = P·e^(rt) gives the value of principal P after t years at continuous rate r; the limit of compounding more and more often.
Compound Interest Formula
A = P(1 + r/n)^(nt) with n compounding periods per year; equivalent to an exponential with base (1 + r/n)^n per year.
Rewriting Exponential Expressions
Using exponent properties, a·b^(x+c) = (a·b^c)·b^x and a·b^(kx) = a·(b^k)^x, so any exponential can be rewritten with a different base or initial value.
Half-Life
The time for a decaying quantity to fall to half its value; the model can be written Q(t) = Q_0·(1/2)^(t/h) where h is the half-life.
Doubling Time
The time for a growing quantity to double; if doubling time is T then Q(t) = Q_0·2^(t/T).
Logarithm
log_b(c) = a means b^a = c; a logarithm is the exponent to which the base must be raised to produce c.
Logarithmic Function
f(x) = a·log_b(x) with b > 0, b ≠ 1, the inverse of the exponential b^x; domain is x > 0 and the graph has a vertical asymptote at x = 0.
Common and Natural Logarithms
log(x) means log_10(x) and ln(x) means log_e(x); both are available on calculators and either can be used with change of base.
log_b(m^k) = k·log_b(m); this lets you bring a variable exponent down to solve exponential equations.
Change-of-Base Formula
log_b(x) = log_a(x)/log_a(b) for any valid base a; often used as ln(x)/ln(b) on a calculator.
Solving Exponential Equations
Isolate the exponential expression, then take a logarithm of both sides and use the power property to bring the exponent down.
Solving Logarithmic Equations
Combine logs into a single log using properties, rewrite in exponential form, solve, and check that every solution keeps all log arguments positive.
Extraneous Solution
A value produced by algebra that makes a logarithm's argument zero or negative in the original equation and must be discarded.
Logarithmic Growth
Logarithmic functions are increasing but concave down; they grow without bound yet more and more slowly, the opposite of exponential growth.
Equal Ratios of Inputs, Equal Differences of Outputs
The defining property of logs: multiplying the input by a constant factor adds a constant to the output, since log(kx) = log(k) + log(x).
Transformations of Log Graphs
g(x) = a·log_b(x − h) + k has vertical asymptote x = h and domain x > h; a < 0 reflects the graph over the x-axis.
Logarithmic Scale
An axis where equal spacing represents multiplication by a constant (powers of 10); used for pH, decibels, and Richter magnitude to display data spanning many orders of magnitude.
Semi-Log Plot
A graph with a logarithmic vertical axis and linear horizontal axis; exponential data appears linear, revealing that y = a·b^x is a good model.
Linearization of Exponential Data
Taking log(y) of exponential data y = a·b^x gives log(y) = log(a) + x·log(b), a line with slope log(b) and intercept log(a).
Exponential Regression
Using technology to fit y = a·b^x to data; appropriate when successive ratios of outputs are roughly constant for equally spaced inputs.
Logistic Function
A model f(x) = c/(1 + a·e^(−bx)) that grows almost exponentially at first, then levels off toward a carrying capacity c; it has two horizontal asymptotes.
Carrying Capacity
The upper limiting value of a logistic model, approached as x → ∞; the growth rate is greatest at the inflection point at half the carrying capacity.
Choosing a Model Family
Constant differences suggest linear, constant second differences suggest quadratic, constant ratios suggest exponential, and a leveling S-shape suggests logistic.
Actual value minus predicted value from a model; a systematic curve in residuals means the model family should be reconsidered.
Newton's Law of Cooling
Temperature difference from surroundings decays exponentially: T(t) = T_s + (T_0 − T_s)·e^(−kt), so the model has a horizontal asymptote at the ambient temperature.