Exponential and Logarithmic Functions

Sequence ● (core concept) — A function from the whole numbers to the real numbers; its graph consists of discrete points, not a continuous curve.

Arithmetic sequence ● (core concept) — A sequence in which successive terms have a common difference d, a constant rate of change: aₙ = a₀ + dn from the initial value a₀, or aₙ = aₖ + d(n − k) from the kth term aₖ.

Geometric sequence ● (core concept) — A sequence in which successive terms have a common ratio r, a constant proportional change: gₙ = g₀·rⁿ from the initial value g₀, or gₙ = gₖ·rⁿ⁻ᵏ from the kth term gₖ. An increasing geometric sequence of positive values grows by larger amounts with each step, whereas an increasing arithmetic sequence grows by equal amounts.

Point-slope form of a linear function ○ — Like arithmetic sequences, linear functions can be written from a known slope m and a point (xᵢ, yᵢ) as f(x) = yᵢ + m(x − xᵢ): an initial value plus repeated addition of a constant rate of change.

Point form of an exponential function ● (core concept) — Like geometric sequences, exponential functions can be written from a known ratio b and a point (xᵢ, yᵢ) as f(x) = yᵢ·bˣ⁻ˣⁱ: an initial value times repeated multiplication by a constant ratio.

Exponential function ● (core concept) — The general form is f(x) = abˣ, with a ≠ 0 (initial value) and base b > 0, b ≠ 1. Its domain is all real numbers. Because outputs are proportional over equal-length input intervals, it is always increasing or always decreasing and always concave up or concave down — so it has no extrema (except on a closed interval) and no points of inflection. Its end behavior is unbounded on one side and approaches zero on the other: lim(x→±∞) abˣ is ∞, −∞, or 0.

Exponential growth and decay ● (core concept) — When a > 0 and b > 1, f(x) = abˣ demonstrates exponential growth; when a > 0 and 0 < b < 1, it demonstrates exponential decay.

Product property of exponents ● (core concept) — bᵐ·bⁿ = bᵐ⁺ⁿ. Graphically, every horizontal translation of an exponential function is equivalent to a vertical dilation: bˣ⁺ᵏ = bᵏ·bˣ.

Power property of exponents ● (core concept) — (bᵐ)ⁿ = bᵐⁿ. Graphically, every horizontal dilation of an exponential function is equivalent to a change of base: bᶜˣ = (bᶜ)ˣ for constant c ≠ 0.

Negative exponent property ● (core concept) — b⁻ⁿ = 1/bⁿ.

Fractional exponent ● (core concept) — An exponential unit fraction b¹/ᵏ, with k a natural number, equals the kth root of b, when it exists.

Growth factor ● (core concept) — In f(x) = abˣ, the base b is the growth factor: the multiplicative change in output per unit change in input, related to a percent change in a contextual scenario. Equivalent forms reveal different properties — for example, 2ᵈ shows a factor of 2 per day, while 2ᵈ = (2⁷)ᵈ/⁷ shows the factor of 2⁷ per week.

Natural base e ● (core concept) — The constant e ≈ 2.718, often used as the base of exponential functions modeling contextual scenarios.

Exponential regression ● (core concept) — An exponential model fitted to a data set with technology.

Residual ● (core concept) — For a given input value, the residual is the actual dependent-variable value minus the value predicted by the regression model.

Residual plot ● (core concept) — A graph of residuals (vertical axis) against the independent variable (horizontal axis). A model is justified as appropriate when its residual plot appears without pattern. Signed residuals, or their absolute values, reveal whether the model underestimates or overestimates actual values on an interval.

Composite function ● (core concept) — If f and g are functions, the composite f ∘ g maps inputs through g and then f: (f ∘ g)(x) = f(g(x)), using the outputs of g as inputs of f. Its domain is restricted to inputs of g whose outputs are in the domain of f.

Composition is not commutative ● (core concept) — f ∘ g and g ∘ f are typically different functions, so f(g(x)) and g(f(x)) are typically different values.

Identity function ● (core concept) — f(x) = x. Composing it with any function g returns g: g(f(x)) = f(g(x)) = g(x); it plays the role of 0 in addition and 1 in multiplication. An additive translation of f can be seen as composition with g(x) = x + k, and a multiplicative dilation as composition with g(x) = kx.

Invertible function ● (core concept) — On a specified domain, a function f is invertible — it has an inverse — when each output value comes from a unique input value. The domain can often be restricted to make a function invertible.

Inverse function ● (core concept) — The reverse mapping f⁻¹: if f(a) = b on f's invertible domain, then f⁻¹(b) = a. The composition of f and f⁻¹ is the identity function: f⁻¹(f(x)) = f(f⁻¹(x)) = x. The domain and range swap roles between f and f⁻¹.

Graph of an inverse function ● (core concept) — The graph of y = f⁻¹(x) is the reflection of the graph of y = f(x) over the line y = x. Analytically, reverse the roles of x and y in y = f(x) and solve for y.

Logarithm ● (core concept) — log_b c is the value the base b must be raised to in order to get c: log_b c = a if and only if bᵃ = c, with b > 0 and b ≠ 1. A logarithm with no written base is understood as the common logarithm, base 10.

Logarithmic scale ● (core concept) — A scale on which each unit represents a multiplicative change by the base of the logarithm: on a base-10 logarithmic scale, units 0, 1, 2, … mark 10⁰, 10¹, 10², ….

Logarithmic function ● (core concept) — The general form is f(x) = a·log_b x, with b > 0, b ≠ 1, and a ≠ 0. It is the inverse of the exponential: log_b x and bˣ undo each other, g(f(x)) = f(g(x)) = x, and the log graph is the reflection of the exponential graph over y = x, with ordered pairs swapped — if (s, t) is an ordered pair of g(x) = bˣ, then (t, s) is an ordered pair of f(x) = log_b x.

Exponential and logarithmic change ● (core concept) — Exponential growth means outputs change multiplicatively as inputs change additively; logarithmic growth is the reverse: outputs change additively as inputs change multiplicatively.

Domain, range, and asymptotes of logarithmic functions ● (core concept) — A general-form logarithmic function has domain x > 0 (all positive real numbers) and range all real numbers. Like exponentials, it is always increasing or always decreasing and always concave up or concave down — no extrema (except on a closed interval), no inflection points. Its graph is vertically asymptotic to x = 0 with unbounded end behavior: lim(x→0⁺) a·log_b x = ±∞ and lim(x→∞) a·log_b x = ±∞.

Product property of logarithms ● (core concept) — log_b(xy) = log_b x + log_b y. Graphically, every horizontal dilation of a log function is a vertical translation: log_b(kx) = log_b k + log_b x.

Power property of logarithms ● (core concept) — log_b(xⁿ) = n·log_b x. Graphically, raising the input to a power is a vertical dilation: log_b(xᵏ) = k·log_b x.

Change of base ● (core concept) — log_b x = (log_a x)/(log_a b) for a > 0, a ≠ 1; all logarithmic functions are vertical dilations of each other.

Natural logarithm ● (core concept) — ln x = log_e x, the logarithm with natural base e.

Extraneous solutions ● (core concept) — Solutions found by analytic or graphical methods must be checked against mathematical or contextual limitations; invalid ones are extraneous — for example, values outside a logarithm's domain.

Logarithmic function model ● (core concept) — Logarithmic functions model situations where input values change proportionally over equal-length output-value intervals (the inverse pattern of exponential growth). If the output value is a whole number, it indicates how many times the initial value has been multiplied by the proportion.

Logarithmic regression ● (core concept) — A logarithmic model fitted to a data set with technology; the natural log function is often useful in real-world models.

Semi-log plot ● (core concept) — A plot with one axis (base n > 1) logarithmically scaled. When the y-axis is log-scaled, data or functions with exponential characteristics appear linear. An advantage: dependent values need not be shifted by a constant to reveal an exponential model.

Linearization of exponential data ● (core concept) — For y = abˣ, the semi-log linear model is log_n y = (log_n b)x + log_n a (n > 0, n ≠ 1): slope log_n b and intercept log_n a. Linear modeling techniques then apply.