1. Exponential/ Logarithmic

2. Exponential Functions
f(x) = ax
Domain (-∞, ∞) Range (0, ∞)
Three types:
1) if 0 < a < 1
2) if a = 1
3) if a > 1

3. Laws of Exponents a x + y = ax ay ax/ ay = a x –y (ax)y = axy
(ab)x = axbx

4. Sketching Example
Sketch the function y = 3 – 2x

5. Exponential Functions are One to One
Has an inverse f-1 which is called the logarithmic function (loga)
f-1(x) = y f(y) = x
ay = x logax = y

6. Example
Find:
log10(0.001)
log216

7. Log Graph Reflection of exponential function about the line y = x
Domain (0, ∞) Range (-∞,∞)

8. Laws of Logarithms loga(xy) = logax + logay loga(x/y) = logax – logay
logaxr = rlogax

9. Example
Evaluate log280 – log25

10. e
y = ax
Many formulas in calculus are greatly simplified if we use a base a such that the slope of the tangent line at y = 1 is exactly 1
For y = 2x, slope at y = 1 is .7
For y = 3x, slope at y = 1 is 1.1
Value of a lies between 2 and 3 and is denoted by the letter e
e =

11. Example
Graph y = ½ e-x – 1 and find the domain and range

12. Natural log (ln)
Log with a base of e
logex = lnx
lnx = y
ey = x

13. Properties of Natural Logs
ln(ex) = x
elnx = x
ln e = 1

14. Example
Find x if lnx = 5

15. Example
Solve e5 – 3x = 10

16. Example
Express ln a + ½ ln b as a single logarithm

17. Expression y = logax ay = x ln ay = ln x y ln a = ln x y = ln x/ ln a
logax = ln x/ ln a if a ≠ 0

18. Example
Evaluate log85

19. Example
The half-life of a radioactive substance given by f(t) = 24 ∙ 2-t/25 Find the inverse