1. Warm-up

2. Logarithmic Functions
8.4
Logarithmic Functions

3. You know that 22 = ________ and 23 = _______
You know that 22 = ________ and 23 = _______. But do you know the value of x that satisfies the equation 2x = 6?
We can guess that the value of x is between ________ and ________, but to find the exact value, we use something called a logarithm.

4. Definition of Logarithm with Base b
If bx = y, then logb y = x.
x = exponent b = base
Read as: log base b of y

5. Example 1: Write the equation in EXPONENTIAL FORM.
a) log3 9 = b) log8 1 = 0
c) log5 1/25 = -2 d) log1/2 2 = -1

6. Example 2: Evaluate each expression.
a) log b) log2 1/8
c) log 1/ d) log32 2

7. Common Logarithm: log with base 10
Notation: log x
If no base is written, assume the base is 10.
Natural Logarithm: log with base e
Notation: ln x

8. Example 3: Use your calculator to evaluate.
a) log b) ln 0.25
c) ln d) log 10

9. Logarithm of 1 Logarithm of base b
Special Logarithm Values- Let b be a positive real number such that b is not 1.
Logarithm of 1
Logb 1 = 0 because b 0 = 1
(Doesn’t matter what your base is)
Logarithm of base b
Logb b = 1 because b 1 = b
Same numbers

10. Finding Inverses Example 5: Find the inverse of each function.
a) y = log4 x b) y = ln (x + 1) c) y = log (x - 3)

11. Teaching Example
Find the inverse of y = logb x Notice: Exponential and Logarithmic functions are inverses!

12. Let f(x) = bx and let g(x) = logb x.
Using Inverses to Simplify Expressions
Let f(x) = bx and let g(x) = logb x.
Verify that these functions are inverses.

13. Properties!
Our verification helped us find properties.
1.)
2.)

14. Example 6: Simplify each expression.
a) 4log4x
b) 10logx
c) 9log9x
d) log327x

15. Warm-up for Day 2
Rewrite in exponential form. 1. Evaluate the expression without a calculator Use a calculator to evaluate. Round to 3 decimal places. 4. ln10 5. log0.3 Simplify 6. 7.

16. Warm-up for Day 3
Rewrite in exponential form. 1. Evaluate the expression without a calculator Use a calculator to evaluate. Round to 3 decimal places. 4. log ln22.5 Simplify 6. 7.

17. Graphing logarithmic functions
The graph of y = logb (x – h) + k has the following characteristics:
x = h is a vertical asymptote
The domain is x > h, the range is all real numbers
If b > 1, the graph moves up and right
If 0 < b < 1, the graph moves down and right

18. Steps to graph Determine the asymptote and sketch it in.
Find convenient points using the un-shifted graph. Make an xy-table
Hint: Always use x = 1 as one of the points.
Make a new xy-table applying the correct shifts.
Left/right changes the x-values
Up/down changes the y-values
Plot new points
Determine domain and range of graph.

19. Example 7: Graph each function. State the domain and range.
a) y = log5 (x + 1) x

20. Example 8: Graph each function. State the domain and range.
b) y = log1/4 (x – 2) – 3 x