1. 3-3 : Functions and their graphs
Lesson objectives
Students will be able to convert equations between logarithmic form and exponential form, evaluate common and natural logarithms, and graph common and natural logarithmic functions.

2. Warm Up
Solve each equation.
8 = x3 2
x1/4 = 2 16
27 = 3x 3
46 = 43x

3. Quick Review

4. Quick Review Solutions

5. What you’ll learn about
Inverses of Exponential Functions
Common Logarithms – Base 10
Natural Logarithms – Base e
Graphs of Logarithmic Functions
… and why
Logarithmic functions are used in many applications, including
the measurement of the relative intensity of sounds.

6. Key Concepts : 𝑓 𝑥 = log 𝑏 𝑥
Logarithm- has base b of a positive number y is defined as follows:
If 𝑦 = 𝑏 𝑥 , then log 𝑏 𝑦 = x.
Common logarithm- a logarithm that uses base 10. ex. log 8

7. Logarithmic Functions are Inverses of Exponential Functions
If 𝑎>0 and 𝑏>0, 𝑏≠ 1, then 𝑦= 𝑙𝑜𝑔 𝑏 𝑥 if and only if 𝑏 𝑦 =𝑥.
Graph:
𝑦 = 𝑏 𝑥
𝑦= 𝑙𝑜𝑔 𝑏 𝑥
𝑦 = 𝑥

9. A. 𝑙𝑜𝑔 2 8 = 3 because B. 𝑙𝑜𝑔 3 3 = 1 2 because D. 𝑙𝑜𝑔 7 7 = 1 because
Evaluating Logarithmic and Exponential Expressions
A. 𝑙𝑜𝑔 2 8 = because
B. 𝑙𝑜𝑔 = because
C. 𝒍𝒐𝒈 𝟓 𝟏 𝟐𝟓 = because
D. 𝑙𝑜𝑔 7 7 = because
E. 6 𝑙𝑜𝑔 = because

10. Logarithmic functions are inverses of exponential functions (x & y are switched)𝑥
𝑓 𝑥 = 2 𝑥
𝑓 𝑥 = 𝑙𝑜𝑔 2 x -2 -1 1 2

11. Common Logarithms, Base 10
Logarithms with base 10 are called common logarithms.
**The subscript 10 is often dropped, so a log statement with no specified base is understood to be base 10.
EX #2: Evaluate the following logarithms and exponential expressions.
A. log 100
C. log
B. log 10
D log 6 6

12. Ex #3: Evaluate these common logarithms with a calculator.
A. log 34.5 B. log 0.43 C. log (−3)

13. Solving Simple Logarithmic Equations
To solve an exponential equation, change it to a logarithmic equation.
To solve a logarithmic equation, change it to an exponential equation.
Ex #4: Solve each equation by changing it to exponential form.
A. log 𝑥=3
B. log 2 𝑥 =5

14. Natural Logarithms, Base e Notation: The logarithmic function 𝐥𝐨𝐠 𝒆 𝒙= 𝐥𝐧 𝒙 .
EX #5: Evaluate the following logarithmic and exponential expressions.
ln 𝑒
ln 𝑒 C. 𝑒 ln 4

15. Evaluating Natural Logarithms with a Calculator
EX #6: Use a calculator to evaluate the logarithmic expressions.
ln = B. ln 0.48
C. ln −5 =

16. Independent Practice 3-3 : 2 to 36 even # - 20 minute

17. Key Concepts
Logarithmic function- the inverse of an exponential function.

18. EX #6: Graph y = log4 x
By definition of logarithm, y = log4 x is the inverse of y = 4x.
Step 1: Graph y = 4x
Step 2: Draw y = x.
Step 3: Choose a few points on 4x. Reverse the coordinates and plot the points of y = log4 x.

19. Ex# 7 :
Graph y = log5 (x – 1) + 2. Step 1: Graph y = log5 x Step 2: Graph the function by shifting the points from the graph to the right 1 unit and up 2 units.

20. Graphing Logarithmic Functions
Basic function: 𝑓 𝑥 = 𝑙𝑛𝑥 or log 𝑥 Graph: Analysis:

21. EX #8: Transforming Logarithmic Graphs
Describe how to transform the graph of 𝑦= ln𝑥 or 𝑦= log𝑥 into the graph of the given function. 𝐴.𝑔 𝑥 =ln⁡(𝑥+2) 𝐵.ℎ 𝑥 =ln⁡(3−𝑥) 𝑐.𝑔 𝑥 = 3log 𝑥 D. ℎ 𝑥 =1+ log 𝑥

22. Independent Practice 3-3 : 38 to 56 even # - 30 minute

23. Home work 3-3 : all
Review 3-3 notes do all problems again.