1. Warm Ups:  Describe (in words) the transformation(s), sketch the graph and give the domain and range:  1) g(x) = e x+4 + 22) y = -(½) x - 3

2. 6.3 Notes: Log basics

3. Exponential Functions:  Exponential functions have the variable located in the exponent spot of an equation/function.  EX: 2 x = 632 x-7 = 987 2x = 54

4. So, what is a logarithm?  Well, if we were given 2 x = 4, we could figure out that x is 2. If we were given 3 x = 27, we could figure out that x = 3. But what about 2 x = 6? Do we know what power 2 is raised to to make 6?  How do we solve this then? Well, just like we would solve any other equation (3x + 7 = 19), we use OPPOSITE OPERATIONS.  The opposite of an exponent is a logarithm

5. Logarithmic form:  The log form is: log b y = x  Translating between forms:  Exponential form:Logarithmic form:  b x = ylog b y = x  “b” is the base  “x” is the exponent  “y” is the “answer”

6. Examples:  Change into log form:  A) 3 x = 9B) 7 x = 343C) 5 x = 625  Change into exponential form:  D) log 6 a = 2E) log 4 16 = yF) log 3 27 = t

7. Common and Natural Logs  The only difference between common logs and natural logs is the base.  The common log has a base of 10. Just like ones, the base of 10 is not written and understood.  Log 10 x = log x  The natural log has a base of “e.” It is not written and understood to be the base.  Log e x = ln x

8. Can we find these answers in the calculator?  ABSOLUTELY! The calculator recognizes only base 10 and base e logarithms. Let’s find the buttons…..  EX: log 8ln 0.3log 15ln 5.72  What do these mean? What are they asking?

9. HW: p. 314 #5 – 16, 27 - 32

10. Warm Up:

11. Inverse properties:  Inverse properties are opposites, they “un-do” each other’s operation.  A) log b b x = xB) = x  EX: log 7 7 4 = =  EX: log 11 11 6 = =  EX: log 5 25 x =

12. Finding Inverse Functions:  Remember, when we found inverse functions before break, we did the following steps:  A) Swap the x and y  B) solve for y using inverse (opposite) operations  C) Simplify the answer if necessary  EX: 1) f(x) = 6 x 2) y = ln(x + 3)  3) h(x) = e x 4) y = log(x + 6)

13. Graphing Log Functions (by hand)

14. Log Graph Basics:  Because the equations of logarithms are inverses (opposites) of exponential equations, the basics of the graphs are also inverses (opposites).  “Go – to” point is (1, 0)  Vertical asymptote at x = 0  To graph by hand, rewrite the log into an exponential equation, make a table of values, then use the inverse of the table (swap the x and y values) to graph the log function.

15. Examples: (don’t forget to give the D & R!)  1) f(x) = log 3 x2) g(x) = log ½ x

16.  Graph:  h(x) = log 5 xa(x) = log ¾ x

17. DUE in CLASS:

18. Warm Up: