# AP Precalculus- Unit 2: Exponential and Logarithmic Functions Flashcards

## Arithmetic and Geometric Sequences

Sequence: an ordered list of numbers, with each listed number being a term. It could be finite or infinite

Arithmetic Sequence: when each successive term in a sequence has a common difference (constant rate of change)

Geometric Sequence: when each successive term in a sequence has a common ratio (consistent proportional change)

Over equal-length input-value intervals, if the output values of a function change:

at a constant rate → linear function (addition)

at a proportional rate → exponential function (multiplication)

^ all can be determined by two distinct sequence or function values

 nth term (arithmetic):an = a0 + dna0 = initial value (zero term)   d = common difference(similar to slope-intercept form) nth term using any term (arithmetic):an = ak + d (n – k)   ak = kth term of the sequence(similar to point-slope form) nth term (geometric):gn = g0rng0 = initial value (zero term)r = common ratio(similar to exponential function) nth term using any term (geometric):gn = gk ∙ r(n – k)gk = kth term of the sequence(similar to shifted exponential)

## Exponential Functions f(x) = bx

-are always increasing or always decreasing

└ no extrema except on a closed interval

-always concave up or always concave down

no inflection points

-if input values increase/decrease without bound, end behavior:

or

or

Horizontal Translation/Vertical Dialation: f(x) = b(x+k) = bx ∙ bk = ab where a = bk

Horizontal Dialation: f(x) = b(cx) → change of the base of function   bc is a constant and c ≠ 0

Exponential functions model growth patterns with successive output values over equal-length input-values intervals are proportional

A constant may need to be added to the dependent variables of a data set to see proportional growth pattern

An exponential function can be constructed from: a ratio and initial value/two input-output pairs

base of exponent (b) → growth factor in successive unite changes in input values; percent change in context

forms of exponential functions can be used in different scenarios:

ex.   if d = number of days            f(x) = 2d → quantity increases by factor of 2 every day

f(x) = (27)(d/7) quantity increase by a factor of 27 every 7 days/week

 General Form of an Exponential Function:An exponential function has the general formf(x) = abxa = initial value ≠ 0b > 0 Additive Transformation of an Exponential Function:g(x) = f(x) + kIf the output values of g are proportional over equal-length input-value intervals, then f(x) is exponential Negative Exponent Property:b-n = (1/bn) Product Property:bmbn = b(m+n) Power Property:(bm)n = bmn The number e:e = 2.718…the natural base e is often used as the base in exponential functions

## Competing Function Model Validation

Model can be an appropriation for data if data set/regression is without pattern

Predicted vs actual results → error in model

May be appropriate to overestimate/underestimate for given interval

## Composition of Functions

(f g)(x)/f(g(x)) à maps set of input values to set of output values such that the output values of g are used as input values of f

└ domain of composite function is restricted to input values of f for which the corresponding output values is the domain of f

Typically, f(g(x)) and g(f(x)) are different values as f g and g f are different functions

Additive Transformations → vertical/horizontal translations (g(x) = x + k)

Multiplicative Transformations → vertical/horizontal dilations (g(x) = kx)

Identity Function:

when f(x) = x

Then g(f(x)) = f(g(x)) = g(x)

Acts as 1 (multiplicative identity) when multiplying

## Inverse Functions

Inverse Function → in each output value is mapped from a unique input value

Ex.  f(x): inputs on x-axis and outputs on y-axis (f(a) = b)  →

f-1(x): inputs on y axis and outputs on x axis (f-1(b) = a)

composite of function and inverse = identity function

function’s domain and range → inverse function’s range and domain (respectively)

*domain may be restricted

## Inverse of Exponential Functions

f(x) = a logb x with base b, where b > 0, b ≠ 1, and a ≠ 0

generally, exponential functions and log functions are inverse functions (reflections over h(x) = x)

└  f(x) = logb x and g(x) = bx   → f (g(x)) = g( f(x)) = x

exponential growth → output values changing multiplicatively as input values change additively

logarithmic growth → output values changing additively as input values change multiplicative

## Logarithmic Functions

logbc = value b must be exponentially raised to in oder to obtain the value c

logbc if and only if ba = c (a & c = constants) (b > 0) (b ≠ 1)

if b not specified, log is common log with base 10 (b = 10)

used to model situations involving proportional growth or repeated multiplication

a logarithmic function can be constructed from a proportion and a real zero/ two input-output pairs

each unit represents a multiplicative change of the base of the log

domain of general form → any real number greater than 0

range of general form → all real numbers

If input values of the additive transformation function g(x) = f (x + k) are proportional over equal-length output value intervals à →(x) is logarithmic  (Does not apply vice versa)

since inverse of exponential function → logarithmic functions are:

- always increasing or always decreasing

- always concave up or always concave down

- do not have extrema except on closed intervals

- do not have points of inflection

In general form, features of log function include:

-vertically asymptotic to x = 0

-end behavior is unbounded

or

 Log Product Property: Log Quotient Property: Log Exponential Property: Natural Log Property:

## Exponential and Logarithmic Equations and Inequalities

- must look for extraneous solutions

-exponential can be written as log functions

combination of transformations of exponential function in general form *

combination of transformation of log function in general form *

* inverse of function can be found by determining the inverse operation to reverse the mapping

## Semi-Log Plots

- y-axis is logarithmically scaled, while the x-axis is linearly scaled

- used to visualize exponential functions

- a constant does not need to be added to reveal that an exponential model is appropriate

- linear model for semi-log plot:

where n > 0 and n ≠ 0

└ linear rate of change:

└ initial linear value: