5. Exponential and Logarithmic Functions

Question 1

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 ______.

Answer 1

b^x

(–∞, ∞)

(0, ∞)

Question 2

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.

Answer 2

x

y = 0

horizontal

Question 3

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 ______.

Answer 3

e

natural

2.72

Question 4

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.

Answer 4

A

P

r

n

Question 5

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.

Answer 5

semiannualy

quaterly

continuous

Question 6

y = log(b)x is equivalent to the exponential form _____, x > 0, b > 0, b ≠ 1.

Answer 6

b^y = x

Question 7

The function f(x) = log(b)x is the ____ function with base _____.

Answer 7

logarithmic

b

Question 8

log(b) b = ____.

Answer 8

1

Question 9

log(b) 1 = _____.

Answer 9

0

Question 10

log(b) b^x = _____.

Answer 10

x

Question 11

b^(log(b) x) = _____.

Answer 11

x

Question 12

Using interval notation, the domain of f(x) = log(b)x is _____ and the range is _____.

Answer 12

(0, ∞)

–∞, ∞

Question 13

The graph of f(x) = log(b)x approaches, but does not touch, the ____-axis.

This axis, whose equation is ____, is a/an _____ asymptote.

Answer 13

y

x = 0

vertical

Question 14

The graph of g(x) = 5 + log(2)x is the graph of f(x) = log(2)x shifted ______.

Answer 14

up 5 units

Question 15

The graph of g(x) = log(3)(x + 5) is the graph of f(x) = log(3)x shifted _____.

Answer 15

to the left 5 units

Question 16

The graph of g(x) = –log(4)x is the graph of f(x) = log(4)x reflected about the _____.

Answer 16

x-axis

Question 17

The graph of g(x) = log(5)(–x) is the graph of f(x) = log(5)x reflected about the _____.

Answer 17

y-axis

Question 18

The domain of g(x) = log(2)(5 – x) can be found by solving the inequality ______.

Answer 18

5 – x > 0

Question 19

The logarithmic function with base 10 is called the _____ logarithmic function.

The function f(x) = log(10)x is usually expressed as f(x) = ______.

Answer 19

common

log x

Question 20

The logarithmic function with base e is called the _____ logarithmic function.
The function f(x) = log(e)x is usually expressed as f(x) = ______.

Answer 20

natural

ln x

Question 21

The product rule for logarithms states that log(b)(MN) = _____.

The logarithm of a product is the _____ of the logarithms.

Answer 21

log(b)M + log(b)N

sum

Question 22

The quotient rule for logarithms states tat log(b)(M/N) = ______.

The logarithm of a quotient is the _____ of the logarithms.

Answer 22

log(b) M – log(b) N

difference

Question 23

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.

Answer 23

p log(b)M

product

Question 24

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 = ____ / _____.

Answer 24

log(a) M / log(a)b

Question 25

If b^M = b^N, then _____.

Answer 25

M = N

Question 26

If 2^(4x – 1) = 2^7, then _____ = 7.

Answer 26

4x – 1

Question 27

If x ln 9 = ln 20, then x = _____.

Answer 27

ln20 / ln9

Question 28

If log(5)(x + 1) = 3, then _____ = x + 1.

Answer 28

5^3

Question 29

If log(3)x + log(3)(x + 1) = 2, then log(3) ______ = 2.

Answer 29

(x² + x)

Question 30

If ln[(7x – 23)/(x + 1)] = ln(x – 3), then ______ = x – 3.

Answer 30

(7x – 23) / (x + 1)

Question 31

True or false:

x^4 = 15 is an exponential equation.

Answer 31

false

Question 32

True or false:

4^x = 15 is an exponential equation.

Answer 32

true

Question 33

True or false:

–3 is a solution of log(5)9 = 2log(5)x.

Answer 33

false

Question 34

True or false:

–10 is a solution of log(5)(x + 35) = 2.

Answer 34

true

Question 35

If e^(0.6x) = 6, then 0.6x = _____.

Answer 35

ln6

Question 36

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.

Answer 36

> 0

< 0

Question 37

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 _____.

Answer 37

A{0}

A

Question 38

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 _____.

Answer 38

A

c

Question 39

y = 3(5)^x can be written in terms of base e as y = 3e^[(____) * x]

Answer 39

ln5