1. Exponential and Logarithmic Functions

2. Exponential Functions Vocabulary – Exponential Function – Logarithmic Function – Base – Inverse Function – Asymptote – Growth – Decay

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

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. 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 x012345 y369121518

7. Inverse Functions Steps for creating an inverse 1.Rewrite the equation from f(x) = to y = 2.Switch variables (letters) x and y 3.Solve equation for y (isolate y again) 4.Rewrite new function as f -1 (x) for new y Example: f(x) = 2x – 3

8. Logarithms Inverse of an exponential function Log b x = y – b is the base (same as exponential function) – Transfers to: b y = x – From exponential function: b x = y Write logarithmic function: log b y = x If there is no base indicated – it is base 10 Example: log x = y

9. Solving & Graphing Logarithms Write out in exponential form: b ? = x What value needs to go in for ? Example: log 3 27 = ? 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 2 x and then log 2 x

10. Properties of Logarithms Product Property: log b x + log b y = log b (x*y) – Example: log 4 8 + log 4 32 Quotient Property: log b x – log b y = log b (x/y) – Example: log 5 75 – log 5 3 Power Property: log b x y = y*log b x – Example: log 2 8 5

11. More Logarithmic Properties Inverse Property: log b b x = x & b log b x = x – Example: log 7 7 5 – Example: 10 log 2 Change of Base: log b x = (log a x ÷ log a b) – Example: log 4 8 – Example: log 5 50

12. Solving Exponentials and Logarithms If the bases of two equal exponential functions are equal – the exponents are equal – Examples: 3 x = 3 2 7 x+2 = 7 2x 4 8x = 16 2 Logarithms are the same: common logarithms with common bases are equal – Examples: log 7 (x+1) = log 7 5 log 3 (2x+2) = log 3 3x Logarithms with logs only on one side – Use the properties of logarithms to solve

13. Logarithmic Equations (Cont) Examples: (Using properties of logarithms) – Log 3 (x – 5) = 2 log 45x – log 3 = 1 – Log 2 x 2 = 8 log x + log (x+9) = 1

14. 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: – 5 x = 7 3 (2x+1) = 15 6 (x+1) + 3 = 12

15. Exponential Inequalities Set up equations the same but use inequality Solve the same as equalities – Example: 2 (n-1) > 2x10 6

16. 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

17. 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

18. 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*e rt A(t), a, r and t represent the same values as previous Example: $750 at 11% compounded continuously

19. Natural Logarithm Inverse of natural base, e Written as ln – Shorthand way to write log e – Properties are the same as for any other log Examples: – ln e 3.2 e ln(x-5) e 2ln x ln e 2x +ln e x Convert between e and ln – e x = 5 ln x = 43

20. Transforming Exponentials

21. Transforming Logarithms