1. MAT 213 Brief Calculus Section 1.3 Exponential and Logarithmic Functions and Models

2. Last time we looked at linear functions What does it mean for a function to be linear? –Constant rate of change –Repeated addition of a quantity What about a situation where we have repeated multiplication? –Say you put $1000 into a bank account paying 5% compounded annually How much would you have after 1 year? How much would you have after 2 years? How much would you have after n years?

3. Exponential Functions f(x) = ab x where a≠0 and b>0 a is the initial quantity (the output corresponding to an input of zero) b is the base or growth ( or decay) factor How does our previous example match up? These functions are called exponential because they have the input variable in the exponent We are repeatedly multiplying by b With linear functions we have a constant change in amount, with exponential functions we have a constant percentage change

4. Exponential Growth f(x)=ab x for b >1 Domain: (-∞,∞)Range: (0,∞) Increasing and Continuous: (-∞,∞) Horizontal Asymptote: x-axis b is a growth factor

5. Now the previous graph was for when b > 1 For an exponential function we only need b > 0 What happens if 0 < b < 1? Can b = 1?

6. Exponential Decay f(x)=ab x for 0 < b <1 Domain: (-∞,∞)Range: (0,∞) Decreasing and Continuous: (-∞,∞) Horizontal Asymptote: x-axis b is a decay factor

7. Exponential Regression Overflow!!! Aligning the data Let’s let 0 correspond to 1990 xf(x)f(x) 1990 1992 1994 1996 1998 2000 6.75 9.12 12.30 16.61 22.42 30.27

8. Doubling time and half-life Doubling time is the amount of time it takes for the output of an increasing exponential function to double Half-life is the amount of time it takes for the output of a decreasing exponential function to decrease by half Let’s find the doubling time for our previous example

9. Logarithmic Functions Definition The logarithmic function log b x is the inverse of the exponential function b x. log b y = x means b x =y EXAMPLES Write as a Logarithm: 5 2 =25 Write as an Exponential function: log 3 9=2

10. Logarithmic Functions On Your Calculator The most frequently used bases are 10 and e LOG calculates log 10 log 10 y = x means 10 x =y LN calculates log e log e y = lny = x means e x = y

11. Below is a graph of both f(x) = ln x and f(x) = e x What is the equation of the dashed line? Describe the behavior of both graphs.

12. Logarithmic Functions f(x)=log b x, a>1 Domain: (0,∞)Range: (-∞,∞) Increasing and Continuous: (0,∞) Vertical Asymptote: y-axis The graph goes through: (1,0)

13. Logarithmic Model If b > 0, we have the increasing, concave down graph we saw in the previous slide What will the graph look like if b < 0? What does a do to our model?

14. #34 from book: The following table gives the average yearly consumption of peaches per person based on that person’s yearly family income when the price of peaches is $1.50 per pound. Yearly income (tens of thousands of dollars) Consumption of peaches (pounds per person per year) 15.0 26.4 37.2 47.8 58.2 68.6

15. Inverses As we saw earlier, exponentials and logs are inverses of each other Just as we saw that the data in the previous slide had a logarithmic look, if we swapped the inputs and outputs, we would have exponential looking data Therefore we can solve for the input of an exponential function using a log and vice versa

16. Composition Property of Inverses If f and g are inverse functions, then This gives way to the following properties involving logs and exponentials Note that the log and the exponential must have the same base!

17. In groups let’s try the following from the book 9, 11, 23