Chapter Three: Exponential and Logarithmic Functions

Question 1

a function has an inverse if and only if

Answer 1

it is a one-to-one function, and therefore passes the horizontal line test

Question 2

the inverse of a function is a reflection over the line

Answer 2

y = x

Question 3

to verify that two functions are inverses of each other

Answer 3

f ° f^-1(x) = x

AND

f^-1 ° f(x) = x

Question 4

exponential function

Answer 4

y = b^x, where b > 1 or 0 < b < 1

Question 5

what are the conditions of an exponential function?

Answer 5

• the base must be a positive real number that DOES NOT equal 1
• the exponent must include a variable
• the exponent can be positive, negative, fractional, or a radical

Question 6

negative power rule

Answer 6

A^-n = (1/A)ⁿ = 1/Aⁿ

Question 7

power of a power rule

Answer 7

(Aⁿ)^m = A^nm

Question 8

fractional power rule

Answer 8

A^m/n = ⁿ√A^m = (ⁿ√A)^m

Question 9

describe the graph of a standard exponential function, where b is greater than one

Answer 9

• the y-intercept is 1 unless the graph has been shifted
• the negative x-axis is a horizontal asymptote
• the function will never be negative (again, unless it has been shifted)
• as b increases, the function on the left will increase more slowly, and the function on the right will increase more quickly

Question 10

what happens when to the graph of an exponential function when the base of the exponential function is a fraction?

Answer 10

the curve is reflected across the y-axis

note: be sure that it is not an improper fraction, as this only holds true when 0 < b < 1

Question 11

what is e?

Answer 11

• equal to 2.71828169
• to estimate e, use the formula ( 1 + 1/n)ⁿ
• the base whose tangent line at (0, 1), or wherever x = 0 has a slop of m = 1

Question 12

what is the compound interest formula? (a.k.a the interest rate formula or present and future value formula)

Answer 12

A = P( 1 + r/m)^mt

Question 13

a logarithm is

Answer 13

the inverse function of an exponential function

Question 14

logarithm to exponential formula

Answer 14

y = log_bx <=> b^y = x

tip: b to what exponent equals x?

Question 15

you can never take the log of

Answer 15

a negative number

Question 16

what are the properties of the graph of a logarithmic function?

Answer 16

• x will always be positive
• the function is asymptotic to the negative y axis
• inverse of exponential graph
• the higher the base, the slower the increase above the x-axis, and the faster the increase below the x-axis (towards the asymptote)

Question 17

a fractional base of a logarithm will affect the graph by

Answer 17

creating a reflection over the x-axis

Question 18

property of logs: when the base and x are the same number

Answer 18

log_bb = 1

Question 19

property of logs: when x equals 1

Answer 19

log_b1 = 0

Question 20

property of logs: multiplying two numbers in the argument of a log, or adding two logs

Answer 20

log_b(xy) = log_bx + log_by

Question 21

property of logs: dividing two logs in the argument, or subtracting two logs

Answer 21

log_b(x/y) = log_bx - log_by

note: this is NOT a quotient of two logs, but the log of a quotient

Question 22

property of logs: the log of a number raised to a power

Answer 22

log_b(x^y) = ylog_bx

Question 23

the natural log

Answer 23

log_ex = lnx

Question 24

the change of base theorem

Answer 24

log_bx = (log_ax)/(log_ab)

a can be any number

Question 25

the richter scale

Answer 25

R = log( I / I₀)

Question 26

the distance modulus formula

Answer 26

M = 5logr - 5

subtract 5 AFTER finding the value of 5logr

base 10 implied

Question 27

what happens when a log is in the exponent, and the base of the log is the same as the base of the exponent?

Answer 27

b^log_bA = A

Question 28

why must you always check your answers when solving logs?

Answer 28

to check for extraneous roots, such as taking the log of a negative number

Question 29

to solve an equation containing logs:

Answer 29

1) combine terms to make one log, and get the constant on one side of the equation, with all logs involving x on the other side
2) use properties of logs
3) convert to exponential form and solve for x

Question 30

the natural log of e is?

Answer 30

1

Question 31

the natural log of a number less than one is?

Answer 31

negative

Question 32

continuously compounding interest formula

Answer 32

A = Pe^rt