1. 8.3 – Logarithmic Functions and Inverses

2. What is a logarithm?
A logarithm is the power to which a number must be raised in order to get some other number
For example, the base ten logarithm of 100 is 2, because ten raised to the power of two is 100:
100 = 102 because log = 2

3. UP, DOWN, UP

4. Remember: If y = bx then logby = x
Ex: Write the following equations in logarithmic form
Remember: If y = bx then logby = x
If 25 = 52
then
Log525=2
If 729 = 36
then
Log3729=6
If 1 = 100
then
Log101=0 If
then

5. Let’s try some: Converting between the two forms.
Expo Form Log Form

6. Let’s pause for a second . . .
If y = bx then logby = x
x in the exponential expression bx is the logarithm in the equation logby=x
The base b in bx is the same as the base b in the logarithm
NOTE: b does not =1 and must be greater than 0
The logarithm of a negative number or zero is undefines.

7. Common Logs
A common log is a logarithm that uses base 10. You can write the common logarithm log10y as log y
Scientists use common logarithms to measure acidity, which increases as the concentration of hydrogen ions in a substance. The pH of a substance equals

8. Evaluating Logarithms
Ex: Evaluate log816
Log816=x Write an equation in log form
16 = 8x Convert to exponential form
24 = (23)x Rewrite using the same base. In this case, base of 2
24 = 23x Power of exponents
4 = 3x Set the exponents equal to each other
x=4/ Solve for x
Therefore, Log816=4/3

9. Evaluating Logarithms
Ex: Evaluate
Write an equation in log form
Convert to exponential form
Rewrite using the same base. In this case, base of 2. Use negative expos!
-5 = 6x Set the exponents equal to each other
x=-5/ Solve for x
Therefore,

10. Let’s try some
Evaluate the following:

11. Let’s try some
Evaluate the following:

12. Graphs of Logarithmic Functions
A logarithmic function is the inverse of an exponential function
In other words, y= 10x and y=log10x are inverses of each other.
Where is the line of reflection?
y = 10x
Y = log10x

13. Let’s try a more complicated one
Find the inverse of y=log5(x-1)+2
y=log5(x -1) Start with the original function
x=log5(y -1) Switch the x and y
x-2=log5(y -1) Subtract 2 from both sides
y-1=5(x-2) Rewrite in y=abx form
y=5(x-2) Add 1 to both sides
The inverse of y=log5(x-1)+2 is y=5(x-2)+1

14. Let’s try some Find the inverse of each function:
Y=log0.5x y=log5x2 y=log(x-2)
Hint: what is the base?

15. Solutions Find the inverse of each function:
Y=log0.5x y=log5x2 y=log(x-2)

16. Now, make a table of values:
How does this help us?
We don’t have a way to graph a log, BUT we can graph an exponential function. If we can find the inverse of the log function and graph it, the graph of the log is simply the reflection of the exponential function.
Let’s start with an easy one: what is the inverse of y=log2x?
y=2x
Now, make a table of values:

17. Graph of y=log2x and y=2x Make a table of values and graph y=2x X y=2x-3
0.125 -2
0.25 -1
0.5 1 2 4 3 8
What is the domain and range of this function?
D: all real numbers. R: y>0

18. Graph of y=log2x and y=2x
Now, reverse the coordinates to graph y=log2x
Prediction: what will it look like? x
y=2x
0.125 -3
0.25 -2
0.5 -1 1 2 4 8 3
What is the domain and range of this function?
D: x>0. R: all real numbers

19. Graph of y=log2x and y=2x Graphing both functions, we get this.
Where is the line of reflection?
y=x

20. Let’s try one:
Graph y=log3(x+3). Determine the domain and range.

21. Let’s try one:
Graph y=log3(x+3)

22. Let’s try one:
Graph y=log3(x+3)

23. Let’s try one:
Graph y=log3(x+3)

24. Determining the domain and range
Domain: x>0
Range: all real numbers
y=log2x
Domain: x>-3
Range: all real numbers
y=log3(x+3)
How can you predict the domain and range without graphing?
Write a sentence summarizing how to determine domain and range.

25. Predict: what is the domain and range of y=log3(x-2) and y=log3(x)+1
y=log3(x-2) y=log3(x)+1
Domain: x>0
Range: all real numbers
Note: the +1 moves the parent graph up but does not affect the left or right translation, so the domain remains x>0
Domain: x>2
Range: all real numbers