1. Exponential and Logarithmic Functions
Unit 3
Exponential and Logarithmic Functions

2. Exponential Functions – Alg 2: 4.1
Vocabulary
Exponential Function y=abx
Base (Common Ratio)- factor of change
Growth (Appreciation)
Decay (Depreciation)
Asymptote – line that a function approaches but never touches
Inverse Function (4.2)
Logarithmic Function ( )
Natural Log/Base e (4.6)

3. Determine Growth vs Decay
Given y=abx
a= y-intercept (starting amount, initial amount)
b= growth or decay factor
If b > 1 = growth
If 0 < b < 1= decay

4. Graphing Exponential Functions
Make a Table of Values
Enter values of X and solve for Y
Plot on Graph
Examples:
Y = 2x state D: and R:
Y = (1\2)x state D: and R:

5. You can model growth or decay by a constant percent increase or decrease with the following formula:
In the formula, the base of the exponential expression, 1 + r, is called the growth factor. Similarly, 1 – r is the decay factor.

6. Appreciation - Growth Amount of function is INCREASING – Growth!
A(t) = a * (1 + r)t
A(t) is final amount
a is starting amount
r is rate of increase
t is number of years (x)
Example: Invest $10,000 at 8% rate – when do you have $15,000 and how much in 5 years?

7. Depreciation - Decay Amount of function is DECREASING – Decay!
A(t) = a * (1 - r)t
A(t) is final amount
a is starting amount
r is rate of decrease
t is number of years (x)
Example: Buy a $20,000 car that depreciates at 12% rate – when is it worth $13,000 and how much is it worth in 8 years?

8. Compounding Interest
Interest is compounded periodically – not just once a year
Formula is similar to appreciation/depreciation
Difference is in identifying the number of periods
A(t) = a ( 1 + r/n)nt
A(t), a and r are same as previous
n is the number of periods in the year

9. Examples of Compounding
You invest $750 at the 11% interest with different compounding periods for 1 yr, 10 yrs and 30 yrs:
11% compounded annually
11% compounded quarterly
11% compounded monthly
11% compounded daily

10. Continuous Compounding
Continuous compounding is done using e
e is called the natural base
Discovering e – compounding interest lab
Equation for continuous compounding
A(t) = a*ert
A(t), a, r and t represent the same values as previous
Example: $750 at 11% compounded continuously

11. Inverse Functions Reflection of function across line x = y
Equivalent to switching x & y values
Example:

12. Switch the x- and y-values in each ordered pair. • • • • •
Example
Graph the relation and connect the points. Then graph the inverse. Identify the domain and range of each relation. x 1 3 4 5 6 y 2
Graph each ordered pair and connect them.
Switch the x- and y-values in each ordered pair. • • x 1 2 3 5 y 4 6 • • •

13. Example Continued
Reflect each point across y = x, and connect them. Make sure the points match those in the table. • • • • • • • • • •
Domain:{1 ≤ x ≤ 6} Range :{0 ≤ y ≤ 5}
Domain:{0 ≤ y ≤5} Range :{1 ≤ x ≤ 6}

14. Inverse Functions Steps for creating an inverse
Rewrite the equation from f(x) = to y =
Switch variables (letters) x and y
Solve equation for y (isolate y again)
Rewrite new function as f-1(x) for new y

15. Inverse Functions Once inverse is set up, use opposite operations
If adding – subtract
If subtracting – add
If multiplying – divide
If dividing – multiply

16. Undo operations in the opposite order of the order of operations.
The reverse order of operations: Addition or Subtraction Multiplication or Division Exponents Parentheses
Helpful Hint

17. Inverse Functions - Examples
Example: f(x) = 2x – 3
Example: f(x)=

18. Example: Retailing Applications
Juan buys a CD online for 20% off the list price. He has to pay $2.50 for shipping. The total charge is $ What is the list price of the CD?
Step 1 Write an equation for the total charge as a function of the list price.
Charge c is a function of list price L.
c = 0.80L

19. c – 2.50 = L 0.80 Example Continued
Step 2 Find the inverse function that models list price as a function of the change.
Subtract 2.50 from both sides.
c – 2.50 = 0.80L
c – 2.50 = L
0.80
Divide to isolate L.
Beware: Don’t change variables (letters) in word problems – they mean or represent something

20. Example Continued
Step 3 Evaluate the inverse function for c = $13.70.
Substitute for c.
L =
13.70 – 2.50
0.80
= 14
The list price of the CD is $14.
Check c = 0.80L
= 0.80(14)
Substitute. =
= 13.70 

21. Logarithms – Alg. 2: Chap 4.3 Inverse of an exponential function
Logbx = y
b is the base (same as exponential function)
Converts to: by = x
From exponential function: bx = y
Write logarithmic function: logby = x
If there is no base indicated – it is base 10
Example: log x = y

22. You can write an exponential equation as a logarithmic equation and vice versa.
Read logb a= x, as “the log base b of a is x.” Notice that the log is the exponent.
Reading Math

23. Example 1: Converting from Exponential to Logarithmic Form
Write each exponential equation in logarithmic form.
Exponential Equation
Logarithmic Form
35 = 243
25 = 5
104 = 10,000
6–1 =
ab = c
The base of the exponent becomes the base of the logarithm.
log3243 = 5 1 2 1 2
log255 =
The exponent is the logarithm.
log1010,000 = 4 1 6
log = –1 1 6
An exponent (or log) can be negative.
logac =b
The log (and the exponent) can be a variable.

24. Example 2: Converting from Logarithmic to Exponential Form
Write each logarithmic form in exponential equation.
Logarithmic Form
Exponential Equation
log99 = 1
log = 9
log82 =
log = –2
logb1 = 0
The base of the logarithm becomes the base of the power.
91 = 9
The logarithm is the exponent.
29 = 512 1 3 1 3
8 = 2
A logarithm can be a negative number. 1 1 16
4–2 = 16
Any nonzero base to the zero power is 1.
b0 = 1

25. Solving & Graphing Logarithms
Write out in exponential form: b? = x
What value needs to go in for ?
Example: log327 = ?
Graphing –
Plot out the Exponential Function – Table of values
Switch the x and y coordinates
Domain of exponential is range of logarithm (limits)
Range of exponential is domain of logarithm (limits)
Example: Plot 2x and then log2x

26. Properties of Logarithms Alg 2: 4.4
Product Property: logbx + logby = logb(x*y)
Example: log64 + log69
Quotient Property: logbx – logby = logb(x/y)
Example: log5100 – log54
Power Property: logbxy = y*logbx
Examples: log log5252

27. Example 1: Product Property
Express log64 + log69 as a single logarithm. Simplify.
log64 + log69
To add the logarithms, multiply the numbers.
log6 (4  9)
log6 36
Simplify. 2
Think: 6? = 36.

28. Example 2: Quotient Property
Express log5100 – log54 as a single logarithm. Simplify, if possible.
log5100 – log5 4
To subtract the logarithms, divide the numbers.
log5(100 ÷ 4)
log525
Simplify. 2
Think: 5? = 25.

29. Example 3 – Power Property
Express as a product. Simplify, if possibly.
a. log104
b. log5252
4log10
2log525
Because = 10, log 10 = 1.
Because = 25, log525 = 2.
4(1) = 4
2(2) = 4

30. More Logarithmic Properties
Inverse Property: logbbx = x & blogbx = x
Example: log775
Example: 10log 2
Change of Base: logbx =
Example: log48
Example: log550
Change of base in Calculator: MATH, logBASE(
log : type base in first then value

31. Example: Geology Application
The tsunami that devastated parts of Asia in December 2004 was spawned by an earthquake with magnitude 9.3 How many times as much energy did this earthquake release compared to the 6.9-magnitude earthquake that struck San Francisco in1989?
The Richter magnitude of an earthquake, M, is related to the energy released in ergs E given by the formula.
Substitute 9.3 for M.

32. Example Continued 13.95 = log 10 E æ ç è ö ÷ ø
Multiply both sides by 3 2
11.8
13.95 = log 10 E æ ç è ö ÷ ø
Simplify.
Apply the Quotient Property of Logarithms.
Apply the Inverse Properties of Logarithms and Exponents.

33. Example Continued
Given the definition of a logarithm, the logarithm is the exponent.
Use a calculator to evaluate.
The magnitude of the tsunami was 5.6  1025 ergs.

34. Apply the Quotient Property of Logarithms.
Example Continued
Substitute 6.9 for M.
Multiply both sides by 3 2
Simplify.
Apply the Quotient Property of Logarithms.

35. Given the definition of a logarithm, the logarithm is the exponent.
Example Continued
Apply the Inverse Properties of Logarithms and Exponents.
Given the definition of a logarithm, the logarithm is the exponent.
Use a calculator to evaluate.
The magnitude of the San Francisco earthquake was 1.4  1022 ergs.
The tsunami released = 4000 times as much energy as the earthquake in San Francisco.
1.4  1022
5.6  1025

36. Solving Exponentials and Logarithms Alg 2: 4.5
If the bases of two equal exponential functions are equal – the exponents are equal
If bases don’t look equal – try and make them (Ex 3)
Examples: 3x = x+2 = 72x x = 16
Examples:
Logarithms are the same: common logarithms with common bases are equal
Examples: log7(x+1) = log log3(2x+2) = log33x

37. Solving Exponentials - continued
Exponents without common bases
Get exponential by itself
Convert to logarithm
Examples:
5x = (2x+1) = (x+1) + 3 = 12

38. Solving Logarithms – continued
Logarithms with logs only on one side
Use the properties of logarithms to solve
Get one log by itself
Must convert to exponential
Examples: (Using properties of logarithms)
Log3(x – 5) = log 45x – log 3 = 1
Log2x2 = log x + log (x+9) = 1

39. Exponential Inequalities
Set up equations the same but use inequality
Solve the same as equalities
Example:

40. Log Inequalities Remember Domain of Log is Only positive x values
Examples:

41. Natural Logarithm Inverse of natural base, e Written as ln
Shorthand way to write loge
Properties are the same as for any other log
Examples: Inverse operations
ln e eln(x-5) e2ln x ln e2x+ln ex
Convert between e and ln
ex = ln x = 43

42. Base e and ln Equations
Examples:
Inequalities Ex:

43. Transforming Exponentials

44. Transforming Logarithms