1. Exponential and Logarithmic Functions

2. Exponential Functions
Vocabulary
Exponential Function
Base (Common Ratio)
Growth (Appreciation)
Decay (Depreciation)
Asymptote
Inverse Function
Logarithmic Function

3. Graphing Exponential Functions
Make a Table of Values
Enter values of X and solve for Y
Plot on Graph
Examples:
Y = 2x
Y = .5x

4. Appreciation 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?

5. Depreciation 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?

6. 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

7. 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

8. 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

9. Inverse Functions Reflection of function across line x = y
Equivalent to switching x & y values
Example:
Inverse Operations
If subtracting – add
If adding – subtract
If multiplying – divide
If dividing – multiply x 1 2 3 4 5 y 6 9 12 15 18

10. Inverse Functions Steps for creating an inverse Example: f(x) = 2x – 3
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
Example: f(x) = 2x – 3

11. Logarithms Inverse of an exponential function Logbx = y
b is the base (same as exponential function)
Transfers 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

12. 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

13. Properties of Logarithms
Product Property: logbx + logby = logb(x*y)
Example: log48 + log432
Quotient Property: logbx – logby = logb(x/y)
Example: log575 – log53
Power Property: logbxy = y*logbx
Example: log285

14. More Logarithmic Properties
Inverse Property: logbbx = x & blogbx = x
Example: log775
Example: 10log 2
Change of Base: logbx = (logax ÷ logab)
Example: log48
Example: log550

15. Natural Logarithm Inverse of natural base, e Written as ln Examples:
Shorthand way to write loge
Properties are the same as for any other log
Examples:
Convert between e and ln
ex = ln x = 43
ln e eln(x-5) e2ln x ln e2x+ln ex
Used for constant (continuous) growth or decay
Example: P 277 – Example 4 : Half life & decay

16. Solving Exponentials and Logarithms
If bases of two equal exponential functions are equal – the exponents are equal
bx = by if and only if x = y
Examples: 3x = x+2 = 72x x = 162
Logarithms are the same: common logarithms with common bases are equal
logbx = logby if and only if x = y
Examples: log7(x+1) = log log3(2x+2) = log33x
Logarithms with logs only on one side
Use the properties of logarithms to solve
Example: log3(x+7) = 3

17. Logarithmic Equations (Cont)
Examples: (Using properties of logarithms)
Log3(x – 5) = log 45x – log 3 = 1
Log2x2 = log x + log (x+9) = 1

18. Solving Logarithms - continued
Exponents without common bases
Use common log to set exponentials equal
Use power property to bring down exponent
Isolate the variable
Divide out the logs – use the calculator
Examples:
5x = (2x+1) = (x+1) + 3 = 12

19. Exponential Inequalities
Set up equations the same but use inequality
Solve the same as equalities
Example: 2(n-1) > 2x106

20. Transforming Exponentials

21. Transforming Logarithms