Unit 2

Exponential and Logarithmic Functions: the one-page sheet

2.1 Change in Arithmetic and Geometric Sequences

Arithmetic sequences add a common difference: a_n = a_k + d(n − k) . Geometric sequences multiply by a common ratio: g_n = g_kr^(n − k) . Equal amounts versus multiplied amounts is the difference that grows into linear versus exponential change.

A sequence is a function whose inputs are whole numbers; its graph is a set of dots.
Arithmetic: common difference $d$ , so $a_n = a_0 + dn$ ; like a line sampled at whole numbers.
Geometric: common ratio $r$ , so $g_n = g_0 r^n$ . An increasing geometric sequence grows by increasing amounts.

an arithmetic sequence, a_n = a_0 + dn , changing by equal amounts a geometric sequence, g_n = g_0r^n , changing by amounts that are multiplied by r use a_n = a_k + d(n − k) or g_n = g_kr^(n − k)

sequence: An ordered list of numbers, called terms, written a_0, a_1, a_2, ... or starting at a_1. It is a function whose inputs are whole numbers (the positions), so its graph is a set of separate dots.
arithmetic sequence: A sequence with a common difference d: each term is the previous term plus d, so a_n = a_0 + dn, or a_n = a_k + d(n - k) from any known term a_k. Its terms change by equal amounts.
geometric sequence: A sequence with a common ratio r: each term is the previous term times r, so g_n = g_0 r^n, or g_n = g_k r^(n - k) from any known term g_k. Its terms change by amounts that are themselves multiplied by r.

2.2 Change in Linear and Exponential Functions

Over equal-length input intervals, a linear function adds a constant difference and an exponential function multiplies by a constant ratio. Justify a model type with both halves and the numbers; with unequal steps, compare rates or per-unit ratios instead.

Linear functions extend arithmetic sequences to all real inputs; exponential functions extend geometric sequences.
Linear: over equal-length input intervals, the outputs change by equal amounts. Exponential: over equal-length input intervals, the outputs change by equal factors (they are proportional).
An increasing exponential function eventually passes any increasing linear function.

only then compare the outputs directly a linear function, f(x) = b + mx an exponential function, f(x) = ab^x , or f(x) = y_1r^(x − x_1) through a known point

exponential function: A function f(x) = ab^x with a not 0, b > 0 and b not 1: the input is in the exponent. Over equal-length input intervals its outputs change by a constant ratio. If b > 1 it shows exponential growth; if 0 < b < 1, exponential decay.

2.3 Exponential Functions

f(x) = ab^x is monotonic with one concavity and a horizontal asymptote, written in limit notation. The base is the growth factor ( 1 + the percent change as a decimal). Adding k moves the asymptote to y = k ; the natural base e is one choice of b .

$f(x) = ab^x$ ( $a \ne 0$ , $b > 0$ , $b \ne 1$ ) is always increasing or always decreasing, has one concavity, and has no extrema or points of inflection.
Its graph has a horizontal asymptote; write the end behavior in limit notation, such as $\lim_{x \to -\infty} 3(2)^x$ $= 0$ .
The base is the growth factor: a $4\%$ increase is $b = 1.04$ ; a $15\%$ decrease is $b = 0.85$ . The natural base $e \approx 2.718$ is a base used in many models.

the function is increasing or decreasing everywhere, and concave up the graph is reflected across the x -axis: concave down one side approaches the asymptote: lim as x → ∞, ab^x = 0 for decay, lim as x → −∞, ab^x = 0 for growth the asymptote moves to y = k

natural base e: The number e, about 2.71828, used as the base of an exponential function, as in f(x) = e^x. It arises naturally when growth happens continuously; a calculator has a key for it.
growth factor: The base b of an exponential function in context: the factor the output is multiplied by for each unit of input. A percent change r (as a decimal) gives the growth factor b = 1 + r: a 4% increase is b = 1.04, and a 15% decrease is the decay factor b = 0.85.

2.4 Exponential Function Manipulation

Exponent rules rewrite exponential functions: b^(x + k) = b^kb^x (a horizontal shift is a vertical dilation), b^cx = (b^c)^x (a horizontal dilation is a new base) and b^(−x) = (1/b)^x . Equivalent forms make different features easy to read.

$b^{x + k} = b^k b^x$ : for an exponential function, shifting the graph horizontally is the same as stretching it vertically.
$b^{cx} = (b^c)^x$ and $b^{x/n} = (b^{1/n})^x$ : a horizontal dilation is the same as changing the base.
Use the rules on whole exponents: $4^{x + 1}$ $= 2^{2(x + 1)}$ $= 2^{2x + 2}$ , never $2^{2x + 1}$ .

by the product rule, b^k⋅b^x : a horizontal translation is a vertical dilation by b^k by the power rule, (b^c)^x : a horizontal dilation is a change of base (1/b)^x : growth reflected across the y -axis is decay

equivalent exponential form: Different ways of writing the same exponential function, found with the exponent rules: 2^(x + 3) = 8(2)^x, 5(4)^x = 5(2)^(2x), (1/3)^x = 3^(-x). A horizontal translation equals a vertical dilation, and a horizontal dilation equals a change of base.

2.5 Exponential Function Context and Data Modeling

Build ab^t from an initial value and a ratio, from two points, or by exponential regression. The base is the growth factor; rewrite it for other time units with rational exponents. Predict and solve with the stored model, and only where the context and data allow.

Two points $(0, a)$ and $(k, y)$ : $b^k = \frac{y}{a}$ , so $b = \left(\frac{y}{a}\right)^{1/k}$ .
Exponential regression on the calculator fits $y = ab^x$ to all the data; store $a$ and $b$ .
Other time units: a yearly factor $b$ is a monthly factor $b^{1/12}$ ; a half-life $h$ gives $\left(\frac{1}{2}\right)^{t/h}$ .

find a and b directly, by the ratio over the gap, or by exponential regression percent change (b − 1)⋅100 per unit; a k -unit factor b^k , a 1/k -unit factor b^(1/k) predict an output, or find the input for a given output with the calculator, inside the domain the context allows

exponential model: A function model of the form f(t) = ab^t, where a is the initial value and b the growth (or decay) factor per unit of t. It is built from an initial value and a ratio, from two data points, or by exponential regression on a calculator.

2.6 Competing Function Model Validation

Residuals (actual minus predicted) measure how far a model misses each data point. Choose the model whose residuals are small and patternless, and read the sign of the residuals to say where a model underestimates or overestimates.

Residual $=$ actual $-$ predicted. Positive: the model underestimates. Negative: it overestimates.
An appropriate model has a residual plot scattered around 0 with no pattern; a U or a trend shows the model has the wrong shape.
Error can also be stated as predicted $-$ actual, or as an absolute error $|\text{predicted} - \text{actual}|$ . Compare competing models by their residuals.

the residuals: positive where the model underestimates, negative where it overestimates a residual plot the model is appropriate; a curve or trend means it misses the shape of the data

residual: Actual value minus predicted value at a data point: positive where a model underestimates, negative where it overestimates. A residual plot graphs the residuals against the input; an appropriate model leaves small residuals with no pattern.

2.7 Composition of Functions

A composite function chains two functions: f(g(x)) puts the output of g into f . Evaluate from the inside out and show the inner value. The order matters, the domain is the inputs whose inner outputs f can take, and the identity function I(x) = x leaves any function unchanged. You can compose from tables, graphs or formulas, and take a function apart into an inner and an outer step.

$f(g(x))$ means: find $g(x)$ first, then put that output into $f$ . Work from the inside out and write the inner value down.
Order matters. $f(g(x))$ and $g(f(x))$ are usually different functions.
The domain of $f(g(x))$ : inputs $x$ in the domain of $g$ whose outputs $g(x)$ are in the domain of $f$ .
The identity function $I(x) = x$ changes nothing: $f(I(x)) = I(f(x)) = f(x)$ , as adding $0$ or multiplying by $1$ .
Composites work in every form: read a table or graph twice, or substitute one formula into the other.

the inner function g turns it into g(x) the output is f(g(x)) , also written (f∘g)(x) the output usually changes: composition is not commutative that x is left out of the domain of f(g(x))

composite function: A function built by using the output of one function as the input of another. The composite $f(g(x))$ , also written $(f \circ g)(x)$ , applies $g$ first and then $f$ .
identity function: The function $I(x) = x$ , which returns every input unchanged. Composing any function with it, in either order, gives the same function back.

2.8 Inverse Functions

An inverse function undoes a function: f⁻¹(b) = a exactly when f(a) = b . It exists when no output repeats, which you justify with specific values, or after restricting the domain. Find it from a table by swapping rows, from a graph by reflecting across y = x , and from a formula by swapping x and y and solving. Composing a function with its inverse gives the identity.

$f^{-1}(b) = a$ means $f(a) = b$ . The $-1$ names the inverse function; $f^{-1}(x)$ is not $\frac{1}{f(x)}$ .
A function is invertible when each output comes from exactly one input (it is one-to-one).
Justify with specific values: " $h(1) = h(3) = 1$ , so $h$ has no inverse", never a bare rule name.
To find a formula, swap $x$ and $y$ in $y = f(x)$ and solve for $y$ .
The domain of $f^{-1}$ is the range of $f$ , and the range of $f^{-1}$ is the domain of $f$ .

f⁻¹ sends b back to a : the point (a, b) becomes (b, a) the graph reflects across the line y = x , and the domain and range trade places the inverse would need two outputs, so h is not invertible that piece has an inverse f(f⁻¹(x)) = x and f⁻¹(f(x)) = x : the identity function

inverse function: The function $f^{-1}$ that undoes $f$ : $f^{-1}(b) = a$ exactly when $f(a) = b$ . It exists when $f$ is one-to-one (each output comes from exactly one input); then $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ .

2.9 Logarithmic Expressions

A logarithm is an exponent: log_b c = a means b^a = c . The common logarithm uses base 10 and the natural logarithm uses base e . Find exact values by writing the input as a power of the base, and estimate others between consecutive powers. Logarithms take only positive inputs. A logarithmic scale spaces numbers by their logarithms, so equal steps are equal factors.

$\log_b c = a$ means $b^a = c$ , for $b > 0$ , $b \ne 1$ , $c > 0$ . A logarithm is an exponent.
Common logarithm: $\log c = \log_{10} c$ . Natural logarithm: $\ln c = \log_e c$ .
Exact values come from writing $c$ as a power of $b$ : $\log_4 8 = \frac{3}{2}$ because $8 = 4^{3/2}$ .
On a logarithmic scale, each equal step multiplies the value by the same factor.

the answer a is log_b c , so log_b c = a means b^a = c you get the common logarithm log c or the natural logarithm ln c log_b c lies between their exponents log_b c exists only for c > 0 , though its value can be negative equal distances mean equal factors: a logarithmic scale

logarithm: The exponent a base needs to produce a number: $\log_b c = a$ means $b^a = c$ (for $b > 0$ , $b \ne 1$ , $c > 0$ ). The common logarithm $\log c$ has base $10$ ; the natural logarithm $\ln c$ has base $e$ .
logarithmic scale: A scale on which numbers are placed by their logarithms, so equal distances stand for equal factors (often factors of $10$ ). It shows values of very different sizes on one axis.

2.10 Inverses of Exponential Functions

With the same base, f(x) = b^x and g(x) = log_b x are inverses: each undoes the other, their graphs reflect across y = x , and their domains, ranges and asymptotes trade places. Adding to the input of an exponential multiplies its output; multiplying the input of a logarithm adds to its output. To invert a stretched exponential, undo the outer steps first, then take the logarithm.

$y = b^x$ and $y = \log_b x$ (same base) are inverse functions: $\log_b(b^x) = x$ and $b^{\log_b x} = x$ for $x > 0$ .
Their graphs reflect across $y = x$ ; $\ln x$ and $e^x$ are the most used pair.
$b^x$ : domain all real numbers, range $y > 0$ , asymptote $y = 0$ . $\log_b x$ : domain $x > 0$ , range all real numbers, asymptote $x = 0$ .
Exponential: equal additions to the input multiply the output. Logarithmic: equal factors on the input add to the output.

g(x) = log_b x sends b^a back to a : they are inverses log_b (b^x) = x for every x , and b^(log_b x) = x for x > 0 the domain (−∞, ∞) and range (0, ∞) of b^x trade places, and y = 0 becomes x = 0 the output is multiplied the output increases by a fixed amount

inverse of an exponential function: An exponential function $y = b^x$ and the logarithm with the same base, $y = \log_b x$ , undo each other: $\log_b(b^x) = x$ for every $x$ and $b^{\log_b x} = x$ for $x > 0$ . Their graphs are reflections across $y = x$ .

2.11 Logarithmic Functions

A logarithmic function f(x) = alog_b x is defined only for x > 0 , with a vertical asymptote at x = 0 . With a > 0 and b > 1 it is increasing and concave down, so its rate of change decreases: lim as x → 0⁺, f(x) = −∞ and lim as x → ∞, f(x) = ∞ , with no extrema. Shifts move the asymptote and the domain. Describe asymptote behavior in limit notation from the side where f is defined.

$f(x) = a\log_b x$ ( $a \ne 0$ , $b > 0$ , $b \ne 1$ ): domain $x > 0$ , range all real numbers, vertical asymptote $x = 0$ .
For $a > 0$ , $b > 1$ : increasing, concave down, $\lim_{x \to 0^+} f(x) = -\infty$ and $\lim_{x \to \infty} f(x) = \infty$ . No extrema, no points of inflection.
Shifts move the asymptote: $\log_b(x - h)$ has domain $x > h$ and asymptote $x = h$ .
Write behavior near the asymptote with a one-sided limit from the side where $f$ is defined.

f(x) = alog_b (x − h) + k has domain x > h and a vertical asymptote x = h for a > 0 and b > 1 , lim as x → h⁺, f(x) = −∞ the outputs grow without bound too, slowly: lim as x → ∞, f(x) = ∞ they decrease, so f is concave down; it is increasing with no maximum or minimum

logarithmic function: A function of the form $f(x) = a\log_b x$ (with $a \ne 0$ , $b > 0$ , $b \ne 1$ ), or a transformation of it. It is defined for $x > 0$ , has the vertical asymptote $x = 0$ , is always increasing or always decreasing, has one concavity and no extrema.

2.12 Logarithmic Function Manipulation

Logarithms turn products into sums, quotients into differences and powers into multiples, because exponents behave that way. Change of base writes any logarithm as a multiple of another, so every logarithmic function is a vertical dilation of every other. On graphs, multiplying the input shifts the graph up or down. Each rule needs positive inputs; watch for domain changes such as ln (x²) versus 2ln x .

$\log_b(xy)$ $= \log_b x + \log_b y$ , $\log_b \frac{x}{y}$ $= \log_b x - \log_b y$ , $\log_b(x^n) = n\log_b x$ , for $x, y > 0$ .
Change of base: $\log_b x = \frac{\ln x}{\ln b}$ . So $\log_4 x = \frac{1}{2}\log_2 x$ , a vertical dilation.
Graphs: $\log_b(kx)$ $= \log_b k + \log_b x$ is a vertical shift (and a horizontal dilation); $\log_b(x^n) = n\log_b x$ is a vertical dilation.
Watch the domain: $\ln(x^2)$ is defined for $x \ne 0$ , but $2\ln x$ only for $x > 0$ .

product property: log_b (xy) = log_b x + log_b y quotient property: log_b x/y = log_b x − log_b y power property: log_b (x^n) = nlog_b x change of base: log_b x = (log_a x)/(log_a b) , so every logarithm is a vertical dilation of every other

product property of logarithms: Rules for rewriting logarithms, for positive $x$ and $y$ : product $\log_b(xy)$ $= \log_b x + \log_b y$ ; quotient $\log_b \frac{x}{y}$ $= \log_b x - \log_b y$ ; power $\log_b(x^n) = n\log_b x$ ; change of base $\log_b x$ $= \frac{\log_a x}{\log_a b}$ .

2.13 Exponential and Logarithmic Equations and Inequalities

To solve an exponential equation, isolate the power and use a common base or a logarithm. To solve a logarithmic equation, combine into one logarithm and rewrite as a power, then check for extraneous solutions. An equation quadratic in e^x is solved with u = e^x , rejecting roots where e^x would not be positive. Inequalities keep their direction for bases above 1 and flip for bases between 0 and 1 .

Exponential equations: isolate $b^{(\ldots)}$ , then use a common base or take $\log$ or $\ln$ of both sides.
Logarithmic equations: get one logarithm on each side or one logarithm equal to a number, then rewrite as a power.
Check every answer in the original equation; a value that makes any logarithm's input $0$ or negative is extraneous.
Inverses of transformed functions: swap $x$ and $y$ , undo the outer steps in reverse order, and use the matching exponential or logarithm.

isolate the power, then take a logarithm of both sides (or use a common base) combine them into one logarithm, then rewrite in exponential form some roots of the new equation are extraneous: check each in the original substitute u = e^x , solve, and reject any u ≤ 0 the direction stays for b > 1 and flips for 0 < b < 1 ; respect the domain

extraneous solution: A value produced while solving that does not satisfy the original equation, often because it makes the input of a logarithm zero or negative, or makes $e^x$ negative. Every candidate must be checked in the original.
exponential equation: An exponential equation has the unknown in an exponent, such as $3 \cdot 2^x = 48$ ; a logarithmic equation has it inside a logarithm, such as $\log_2(x + 1) = 3$ . Each is solved by undoing one function with its inverse.

2.14 Logarithmic Function Context and Data Modeling

Use a logarithmic model y = a + bln x when the input must multiply by a constant for the output to rise by a constant. Build it from two points or by logarithmic regression, keeping the constants stored. Predict outputs by substituting and find inputs by rewriting as a power. The model is concave down, so its rate of change decreases; it is supported only near the data and fails near x = 0 .

A logarithmic model $y = a + b\ln x$ fits when equal factors in $x$ give equal changes in $y$ .
Build it from two points (one with $x = 1$ makes $a$ easy) or with logarithmic regression on a calculator.
$b > 0$ : increasing, concave down, so the rate of change decreases: growth that slows but never stops.
Justify the choice with the numbers: "each time $x$ triples, $y$ increases by $6$ ".

the data fit y = a + blog_c x , a logarithmic model the logarithm is 0 there, so the output at x = 1 is a use logarithmic regression, y = a + bln x , and store the constants isolate the logarithm and rewrite as a power the model is unsupported or gives impossible values, so restrict the domain

logarithmic model: A function model of the form $y = a + b\ln x$ (or with another base), used when equal factors in the input go with equal changes in the output. It is built from two points or by logarithmic regression.

2.15 Semi-log Plots

A semi-log plot puts one axis on a logarithmic scale. Exponential data y = a⋅b^x then fall on a straight line, because log y = log a + xlog b . The slope of the line is log b and its intercept is log a , so you can read the model's growth factor and starting value from the line. Linear data, or exponential data with a vertical shift, bend instead.

A semi-log plot has one axis (usually the vertical one) on a logarithmic scale with some base $n > 1$ .
Exponential data $y = a \cdot b^x$ look linear on it, because $\log_n y$ $= \log_n a + x\log_n b$ .
From the line: $b = n^{\text{slope}}$ and $a = n^{\text{intercept}}$ . With a natural-log scale, use $e$ .
A function with a vertical shift, $a \cdot b^x + k$ with $k \ne 0$ , does not give a line.

log y = log a + xlog b , a linear function of x the data lie on a straight line: a semi-log plot slope = log b and intercept = log a , so b = 10^slope and a = 10^intercept they are not exponential of the form a⋅b^x (linear data bend down; a vertical shift bends too)

semi-log plot: A graph with one axis, usually the vertical one, on a logarithmic scale. Exponential data $y = a \cdot b^x$ appear as a straight line on it, with slope $\log b$ and intercept $\log a$ .