Chpt. 7, Exponential and Logarithmic Functions Flashcards Preview

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Flashcards in Chpt. 7, Exponential and Logarithmic Functions Deck (21)
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1
Q

exponential function

A

The general function of an exponential function is y = ab^x, where x is a real number, a does not equal 0, and b does not equal 1. When b is greater than 1, the function models exponential growth with growth factor b. When b is between 1 and 0, the function models exponential decay with decay factor b.

2
Q

exponential growth

A

Modeled by an exponential function when b is greater than 1.

3
Q

exponential decay

A

Modeled by an exponential function when b is between 1 and 0.

4
Q

asymptote

A

A line that a graph approaches as x or y increases in absolute value.

5
Q

growth factor

A

The variable “b,” when in an exponential function that models exponential growth.

6
Q

decay factor

A

The variable “b” when in an exponential function that models exponential decay.

7
Q

function to model exponential growth or decay

A

A(t) = a(1 + r)^t, where:

A = amount after t time periods (or F)
T = number of time periods
a = initial amount
r = rate of growth/decay
8
Q

natural base exponential function

A

An exponential function with base e. E = 2.71828(infinity). It is a number similar in usage to pi that becomes very important in PreCalc. and Calc.

9
Q

continuously compounded interest

A

When interest is compounded continuously on principal P, the value A of an account is A = PE^rt.

10
Q

exponential function transformations, from parent:

y = b^x; a = 1

to:

  1. stretch
  2. compression
  3. reflection in x-axis
  4. horizontal translation
  5. vertical translation
  6. all transformations combined
A
  1. y = ab^x; abs. val. of a is > 1
  2. y = ab^x; abs. val. of a is 0
  3. ab^x; a
11
Q

logarithm

A

The logarithm base b of a positive number x is defined as log(b)x = y, if and only if x = b^y.

12
Q

logarithmic function

A

The inverse of an exponential function.

13
Q

common logarithm

A

A logarithm that uses base 10, and thus are written as log(10)x = y. When logx = y is written, and there is no base for the log, it is assumed that the base is 10.

14
Q

logarithmic scale

A

A scale that uses the logarithm of a quantity, rather than the quantity itself.

15
Q

logarithmic function transformations, from parent:

y = log(b)_x(); b > 0; b does not equal 1

to:

  1. stretch
  2. compression
  3. reflection in x-axis
  4. horizontal translation
  5. vertical translation
  6. all transformations combined
A
  1. y = (a)log(b)_(x); abs. val. of a is > 1
  2. y = (a)log(b)_(x); abs. val. of a is between 1 and 0
  3. y = (a)log(b)_x; a is
16
Q

change of base formula

A

log(b)_(M) =

(log(c)(M)) / (log(c)(b)),

where M, b, and c are all positive numbers, are where neither b nor c equal 1

17
Q

properties of logarithms:

  1. product property
  2. quotient property
  3. power property
A
  1. log(b)(mn) = log(b)(m) + log(b)_(n)
  2. log(b)(m/n) = log(b)(m) - log(b)_n)
  3. log(b)(m^n) = (n)log(b)(m)

*The above properties apply only when m, n, and b are all positive, and when b does not equal 1.

18
Q

exponential equation

A

This type of equation is in the form y = b^cx, in which the variable is located in the exponent.

19
Q

logarithmic equation

A

An equation that includes a logarithm involving a variable. For example, log(3)_(x) = 4.

20
Q

natural logarithmic function

A

A logarithmic function with base e. The natural logarithmic function, log(E)_(x), can also be written as y = ln(x). It is the inverse of y = e^x. Ln is basically another way of writing log(e).

21
Q

more properties of logarithms;

b^0 = 1, so log(b)(1) =
b^1 = b, so log(b)
(b) =
b( log(b)(x) ) = x, so log(b)(b^x) =
b^x(b^y) = b^(x +y), so log(b)_(x/y) =

A

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