2. Exponential Function An exponential function with base b and exponent x is defined by Ex. Domain: All reals Range: y > 0 (0,1) 0 1 1 3 2 9 x y

3. Laws of Exponents LawExample

4. Properties of Exponential Functions 1.The domain is. 2. The range is (0, ). 3. It passes through (0, 1). 4. It is continuous everywhere. 5. If b > 1 it is increasing on. If b < 1 it is decreasing on.

5. Examples Ex.Simplify the expression. Ex.Solve the equation

6. Logarithms An logarithmic of x to the base b is defined by Ex.

7. Examples Ex. Solve each equation a. b.

8. Properties of Logarithms log a 1 = 0 log a a = 1 log a a x = x If log a x= log a ythen x = y because a 0 = 1 because a 1 = a Change-of-Base Product Property Quotient Property Power Property

9. Notation Common Logarithm Natural Logarithm

10. Product Property = log a M + log a N1) log a MN = log b A + log b T2) log b AT = log M + log A + log T + log H3) log MATH Express as a sum of logarithms.

11. Express as a single logarithm = log 5 (19*3) Ex. 4) log 5 19 + log 5 3 5) log C + log A + log B + log I + log N = log CABIN

12. Express as a sum of logarithms, then simplify 6) log 2 (4*16)= log 2 4 + log 2 16 = 2 + 4 = 6

13. Use log 5 3 = 0.683 and log 5 7 = 1.209 to approximate… log 5 (21) = log 5 3 + log 5 7 = 0.683 + 1.209 = 1.892 = log 5 (3*7)

14. Example Use the laws of logarithms to simplify the expression:

15. Expand Simplify the division. Simplify the multiplication of 4  Change the radical sign to an exponent Express the exponent as a product

16. Condense Express all products as exponents Simplify the subtraction. Change the fractional exponent to a radical sign. Simplify the addition.

18. Logarithmic Function An logarithmic function of x to the base b is defined by Properties: LOG function is inverse to exponential function 1. Domain: (0, ) 2.Range: 3. Intercept: (1, 0) 4. Continuous on (0, ) 5. Increasing on (0, ) if b > 1 Decreasing on (0, ) if b < 1

19. Logarithmic Function Graphs Ex. (1,0)

20. Ex. Solve Apply ln to both sides.