1. Logarithms and Logarithmic Functions
Unit 6.6

2. Warm up…
The Richter scale describes the intensity of an earthquake. It was developed by Charles Richter in The table shows how the intensity of an earthquake increases as the number increases.
Use a graphing
calculator to make a scatter plot for the data in the table. Sketch the graph in your notes including the window.
2. Find an exponential equation to fit this data.

3. Warm up… 1. 2.

4. Reflecting Exponential Functions
Work and complete the worksheet when you have finished your quiz.

5. How do I solve x = 3y ?
John Napier was a Scottish theologian and mathematician who lived between 1550 and He spent his entire life seeking knowledge, and working to devise better ways of doing everything from growing crops to performing mathematical calculations. He invented a new procedure for making calculations with exponents easier by using what he called logarithms. A logarithm can be written as a function y = logbx.
The notation y = logbx is another way of writing x = by.
So x = by and y = logbx represent the same functions.
y =log3x is simply another way of writing x = 3y.
The notation is read “y is equal to the logarithm, base 3, of x.”

6. Logarithmic Function (Common)
Calculator: y1= log(x)
Domain: x > 0 or x  (0,∞)
Range: y   or y  (-∞,∞)
Zeros: (1,0) or x = 1
X-Intercept: (1,0)
Y-Intercept: none

7. Logarithmic Function (Common)
Symmetry: None
Max: None
Min: None
Increasing: x  (0,∞)
Decreasing: Never
Vertical Asymptotes: x = 0
Horizontal Asymptotes: None

8. Logarithmic Function (Natural)
Calculator: y1= ln(x)
Domain: x > 0 or x  (0,∞)
Range: y   or y  (-∞,∞)
Zeros: (1,0) or x = 1
X-Intercept: (1,0)
Y-Intercept: none

9. Logarithmic Function (Natural)
Symmetry: None
Max: None
Min: None
Increasing: x  (0,∞)
Decreasing: Never
Vertical Asymptotes: x = 0
Horizontal Asymptotes: None

10. Things I should Know about Logarithmic Functions
The output of a log function is an exponent.
Log and exp. functions are inverses.
Domain: (0, )
Range: (-, )
x-intercept of the graph: (1,0)
Vertical Asymptote at x = 0 (or the y-axis)

11. Definition of Logarithm
Exponential Form Logarithmic Form
(base)(exp) = (product) (exp) = log(base)(product)
EX: y = y = log525.
THE QUESTION that you’re trying to answer:
What exponent, y, takes a base of b to a product of x?
EX: What exponent, y, takes a base of 5 to 25?
y = 2

12. Example 1….
Change from Log notation to Exponential notation or Exponential notation to Log notation.
Logarithmic Form Exponential Form
(___ is the exp. that takes a base of __ to a __.) 2 3 9
(___ is the exp. that takes a base of __ to an __.) 3 x 8

13. Example 2….
Change from Log notation to Exponential notation or Exponential notation to Log notation.
Logarithmic Form Exponential Form
(__ is the exp. that takes a base of __ to an ___.) 4 3 81

14. Only One Graph Is Visible.
Equivalent Equations
x = 3y and y = log3x
Only One Graph Is Visible.

15. Two graphs are visible that are reflected over the y = x line.
Inverse Equations
The inverse of the exponential parent function can be defined as a new function, the logarithmic parent function. The functions are reflections of each other over the line y = x.
y = 2x and y = log2x
Two graphs are visible that are reflected over the y = x line.

16. Finding the inverse algebraically
Example 3: Rewrite the inverse of the exponential function y = 2x + 3 as a logarithmic function.
Beginning Equation
Replace x with y and y with x.
Isolate the term containing y.
Rewrite the exponential function as a logarithmic function

17. Finding the inverse remember…
When using a table of values to find the inverse of an exponential function the domain will switch with the range like on the beginning activity.
When using a graph to find the inverse of an exponential function, the graph must reflect over the y = x line.
When using algebraic methods to find the inverse of an exponential function, switch x and y in the equation, get the term containing y by itself, and then rewrite in logarithmic form.

18. Example 4 y = 4x-2. log2x+3 x = 4y-2. x = log2y+3 x+2 = 4y x-3 = log2y
Determine the inverse of the following. Your final answer should be in the equation box below.
y = 4x-2.
log2x+3
x = 4y-2.
x = log2y+3
x+2 = 4y
x-3 = log2y
y=log(x+2)
y=2x-3

19. Example 4 Determine the inverse of the following. y = 6x-5
y = log6(x+2)
x = 6y-5
x = log6(y+2)
log6x= y-5
6x = y+2
log6x+5=y Or
y = log6x+5
y=6x - 2

20. Example 5…. Evaluate the expression. Set equal to x
Re-write in exponential form.
Solve for x.

21. Example 5…
Definition of Logarithms

22. Example 6… Definition of Logarithms Property of Equality
Equate the Exponents
Solve for x

23. Cont…
The solution checks!

24. Example 7… Definition of Logarithms Power of Power
Equate the Exponents
Check your solution!

25. Example 7…
The solution checks!

26. Forms: Logarithmic Functions
Parent Function:
Common Log:
Natural Log:
Standard Form:
Transfm Form:

27. Absent Students-Notes 6.6
Attach this note page into your notebook
Complete all examples and warm-ups
Be sure to understand the difference between finding the inverse, evaluating, solving and rewriting logs and exponentials