1. Definition
y=log base a of x
if and only if

2. Important Idea
Logarithmic Form
Exponential Form

3. Important Idea
The logarithmic function is the inverse of the exponential function

4. Important Idea
In your book and on the calculator, is the same as If no base is stated, it is understood that the base is 10.

5. Try This Without using your calculator, find each value: 5 1 1/3
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6. Example
Solve each equation by using an equivalent statement:

7. Definition
A second type of logarithm exists, called the natural logarithm and written ln x, that uses the number e as a base instead of the number 10. The natural logarithm is very useful in science and engineering.

8. Important Idea
Like , the number e is a very important number in mathematics.

9. Important Idea The natural logarithm is a logarithm with the base e
is a short way of writing:

10. Definition
If and only if

11. Try This
Use a calculator to find the following value to the nearest ten-thousandth:
1.1394

12. Try This
Solve each equation by using an equivalent statement:
x=7.389
x=2.079

13. Example Using your calculator, graph the following:
Where does the graph cross the x-axis?

14. Example Using your calculator, graph the following:
Can ln x ever be 0 or negative?

15. Example Using your calculator, graph the following:
What is the domain and range of ln x?

16. Example Using your calculator, graph the following:
How fast does ln x grow? Find the ln 1,000,000.

17. Try This Using your calculator, graph:
Describe the differences. How does the domain and range change?

19. Try This
Solve for x:
1.151
-.077
531434
-2 , -1

20. Try This
Solve for x:
2.944
.564 6

21. Important Idea
The definitions of common and natural logarithms differ only in their bases, therefore, they share the same properties and laws.

22. Important Idea Properties of Common Logarithms:
log x defined only for x>0
log 1=0 & log 10=1
for x >0

23. Important Idea Properties of Natural Logarithms:
ln x defined only for x>0
ln 1=0 & ln e=1
for x >0

24. Important Idea

25. MUST REMEMBER ln an=n ln a Product Law: ln(ab)=ln a + ln b
Quotient Law:
Power Law:
ln an=n ln a

26. Same Rules for any base
Product Law:
Quotient Law:
Power Law:

27. Use a combination of logarithmic properties and laws to re-write the given expression:

28. Express
In terms of log A, log B, and
Log C