Exponential / Logarithmic Functions

Question 1

What’s the base exponential function?

Answer 1

f(x) = ab^x

a is the initial value, b is the base where b > 0 and not equal to 1

Question 2

Describe how the values of the base affect the graph

Answer 2

If b > 1, increasing function/exponential growth
If 0 < b < 1, decreasing function/decay

Question 3

What are the other features of a base exponential function?

Answer 3

y-int: (0, a)
x-int: none
D: real numbers
R: y > 0
H.A: y = 0
V.A: none

Question 4

What is the constant ratio?

Answer 4

The ratio between consecutive y-values when x increases by 1

Question 5

What are some other ways to find b?

Answer 5

b = y2/y1

If the y-value is known at x = 1, then b = y-value / a

Question 6

How do you graph an exponential function?

Answer 6

At least 3 points needed, one being the y-int.

Question 7

What is the general equation of a transformed exponential function?

Answer 7

f(x) = ab^(k(x - d)) + c

Question 8

What are 2 methods to solve an exponential equation?

Answer 8

1. Change the bases to match
2. Create a factorable equation using b^x or other

Question 9

What is a logarithm?

Answer 9

It’s the inverse of an exponential function

Question 10

Explain how a logarithm works

Answer 10

If b^x = y, then log (base b) y = x
It asks what power of b gives y

Question 11

Compare exponential form to log form

Answer 11

Exponential: Domain real numbers, range y > 0, x-int (0, 1), x-int none, H.A y = 0

Log form: Domain x > 0, range real numbers, y-int none, x-int (1, 0), V.A x= 0

Question 12

Convert 2^5 = 32 to log form

Answer 12

5 = log (base 2) 32

Question 13

Are logs one-to-one functions?

Answer 13

Yes

Question 14

State the 4 log properties

Answer 14

1. log (base a) 1 = 0
2. log (base a) a = 1
3. log (base a) a^x = x
4. a^(log (base a) x) = x

Question 15

Log law: log (base a) xy is equal to what?

Answer 15

log (base a) x + log (base a) y

Question 16

Log law: log (base a) x/y is equal to what?

Answer 16

log (base a) x - log (base a) y

Question 17

Log law: log (base a) x^k is equal to what?

Answer 17

klog (base a) x

Question 18

Change of base law: log (base a) b is equal to what?

Answer 18

log (base m) b divided by log (base m) a

Question 19

How do you simplify log (base a^n) b?

Answer 19

Use change of base law

Question 20

When is log (base a) x defined?

Answer 20

When x > 0 since you can only take the log of a positive number

Question 21

Do you need domain restrictions for logs?

Answer 21

Yes

Question 22

What is Euleur’s number?

Answer 22

e (2.7182…)

Question 23

What is a natural log?

Answer 23

It’s log (base e) x, written as lnx

Question 24

To graph a lnx function, what are the steps?

Answer 24

1. Start w/ e^x, then lnx, then continue w/ transformations
2. Find y-int and x-int
3. Graph it

Question 25

For growth and decay, what is the general formula?

Answer 25

A = k times c^(t/d) A is final value, k is starting value, t is time elapsed, d is time per event, c is the base

Question 26

The base c can have multiple values, describe them

Answer 26

2 - doubling 3 - tripling 1/2 - half life 0.8 - depreciate by 20% 1.05 - 5% increase

Question 27

For growth and decay, what is the other general formula?

Answer 27

A = Pe^kt A is the final value, P is starting value, t is time, k is modifier. If k > 0, growth. If k < 0, decay.

Question 28

What is the general formula for compound interest?

Answer 28

A = P(1 + i/n)^nt A is the final value, P is principle value, t is time, i is the interest rate per annum, n is the number of compounding periods