Chpt. 7, Exponential and Logarithmic Functions Flashcards
exponential function
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.
exponential growth
Modeled by an exponential function when b is greater than 1.
exponential decay
Modeled by an exponential function when b is between 1 and 0.
asymptote
A line that a graph approaches as x or y increases in absolute value.
growth factor
The variable “b,” when in an exponential function that models exponential growth.
decay factor
The variable “b” when in an exponential function that models exponential decay.
function to model exponential growth or decay
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
natural base exponential function
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.
continuously compounded interest
When interest is compounded continuously on principal P, the value A of an account is A = PE^rt.
exponential function transformations, from parent:
y = b^x; a = 1
to:
- stretch
- compression
- reflection in x-axis
- horizontal translation
- vertical translation
- all transformations combined
- y = ab^x; abs. val. of a is > 1
- y = ab^x; abs. val. of a is 0
- ab^x; a
logarithm
The logarithm base b of a positive number x is defined as log(b)x = y, if and only if x = b^y.
logarithmic function
The inverse of an exponential function.
common logarithm
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.
logarithmic scale
A scale that uses the logarithm of a quantity, rather than the quantity itself.
logarithmic function transformations, from parent:
y = log(b)_x(); b > 0; b does not equal 1
to:
- stretch
- compression
- reflection in x-axis
- horizontal translation
- vertical translation
- all transformations combined
- y = (a)log(b)_(x); abs. val. of a is > 1
- y = (a)log(b)_(x); abs. val. of a is between 1 and 0
- y = (a)log(b)_x; a is
change of base formula
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
properties of logarithms:
- product property
- quotient property
- power property
- log(b)(mn) = log(b)(m) + log(b)_(n)
- log(b)(m/n) = log(b)(m) - log(b)_n)
- 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.
exponential equation
This type of equation is in the form y = b^cx, in which the variable is located in the exponent.
logarithmic equation
An equation that includes a logarithm involving a variable. For example, log(3)_(x) = 4.
natural logarithmic function
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).
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) =
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