5. Exponential and Logarithmic Functions Flashcards
The exponential function f with base b is defined by f(x) = _____, b > 0 and b ≠ 1.
Using interval notation, the domain of this function is _____ and the range is ______.
b^x
(–∞, ∞)
(0, ∞)
The graph of the exponential function f with base b approaches, but does not touch, the ____-axis.
This axis, whose equation is _____, is a/an ______ asymptote.
x
y = 0
horizontal
The value that (1 + 1/n)^n approaches as n gets larger and larger is the irrational number _____, called the ______ base.
This irrational number is approximately equal to ______.
e
natural
2.72
Consider the compound interest formula A = P(1 + r/n)^nt.
This formula gives the balance, _____, in an account with principal _____ and annual interest rate ______, in decimal form, subject to compound interest paid _____ times per year.
A
P
r
n
If compound interest is paid twice a year, we say that the interest is compounded _____.
If compound interest is paid four times a year, we say that the interest is compounded ______.
If the number of compounding periods increases infinitely, we call this ______ compounding.
semiannualy
quaterly
continuous
y = log(b)x is equivalent to the exponential form _____, x > 0, b > 0, b ≠ 1.
b^y = x
The function f(x) = log(b)x is the ____ function with base _____.
logarithmic
b
log(b) b = ____.
1
log(b) 1 = _____.
0
log(b) b^x = _____.
x
b^(log(b) x) = _____.
x
Using interval notation, the domain of f(x) = log(b)x is _____ and the range is _____.
(0, ∞)
–∞, ∞
The graph of f(x) = log(b)x approaches, but does not touch, the ____-axis.
This axis, whose equation is ____, is a/an _____ asymptote.
y
x = 0
vertical
The graph of g(x) = 5 + log(2)x is the graph of f(x) = log(2)x shifted ______.
up 5 units
The graph of g(x) = log(3)(x + 5) is the graph of f(x) = log(3)x shifted _____.
to the left 5 units
The graph of g(x) = –log(4)x is the graph of f(x) = log(4)x reflected about the _____.
x-axis
The graph of g(x) = log(5)(–x) is the graph of f(x) = log(5)x reflected about the _____.
y-axis
The domain of g(x) = log(2)(5 – x) can be found by solving the inequality ______.
5 – x > 0
The logarithmic function with base 10 is called the _____ logarithmic function.
The function f(x) = log(10)x is usually expressed as f(x) = ______.
common
log x
The logarithmic function with base e is called the _____ logarithmic function.
The function f(x) = log(e)x is usually expressed as f(x) = ______.
natural
ln x
The product rule for logarithms states that log(b)(MN) = _____.
The logarithm of a product is the _____ of the logarithms.
log(b)M + log(b)N
sum
The quotient rule for logarithms states tat log(b)(M/N) = ______.
The logarithm of a quotient is the _____ of the logarithms.
log(b) M – log(b) N
difference
The power rule for logarithms states that log(b)M^p = ______.
The logarithm of a number with an exponent is the ______ of the exponent and the logarithm of that number.
p log(b)M
product
The change-of-base property allows us to write logarithms with base b in terms of a new base a. Introducing base a, the property states that
log(b)M = ____ / _____.
log(a) M / log(a)b
If b^M = b^N, then _____.
M = N
If 2^(4x – 1) = 2^7, then _____ = 7.
4x – 1
If x ln 9 = ln 20, then x = _____.
ln20 / ln9
If log(5)(x + 1) = 3, then _____ = x + 1.
5^3
If log(3)x + log(3)(x + 1) = 2, then log(3) ______ = 2.
(x² + x)
If ln[(7x – 23)/(x + 1)] = ln(x – 3), then ______ = x – 3.
(7x – 23) / (x + 1)
True or false:
x^4 = 15 is an exponential equation.
false
True or false:
4^x = 15 is an exponential equation.
true
True or false:
–3 is a solution of log(5)9 = 2log(5)x.
false
True or false:
–10 is a solution of log(5)(x + 35) = 2.
true
If e^(0.6x) = 6, then 0.6x = _____.
ln6
Consider the model for exponential growth or decay given by
A = A{0}e^(kt).
If k ____, the function models the amount, or size, of a growing entity.
If k _____, the function models the amount, or size, of a decaying entity.
> 0
< 0
In the model for exponential growth or decay, the amount, or size, at t = 0 is represented by _____.
The amount, or size, at time t is represented by _____.
A{0}
A
Consider the model for limited logistic growth given by
A = c / [1 + ae^(–bt)].
The amount, or size, at time t is represented by _____.
This value can never exceed _____.
A
c
y = 3(5)^x can be written in terms of base e as y = 3e^[(____) * x]
ln5
