Logarithms and

Logarithmic Functions

Unit 6.6

Warm up…

1. 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.

The Richter scale describes the intensity of an earthquake. It was developed by Charles Richter in 1935. The table shows how the intensity of an earthquake increases as the number increases.

Warm up…1.

1 1010

xy 2.

Reflecting Exponential Functions

Work and complete the worksheet when you have finished your quiz.

How do I solve x = 3y ? John Napier was a Scottish theologian and

mathematician who lived between 1550 and 1617. 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.”

Logarithmic Function(Common)

10( ) logf x x1; 10b b

Calculator: y1= log(x)Domain: x > 0 or x (0,∞)Range: y or y (-∞,∞)Zeros: (1,0) or x = 1X-Intercept: (1,0)Y-Intercept: none

Logarithmic Function(Common)

10( ) logf x x1; 10b b

Symmetry: NoneMax: NoneMin: None Increasing: x (0,∞)Decreasing: NeverVertical Asymptotes: x = 0Horizontal Asymptotes: None

Logarithmic Function (Natural)

( ) lnf x x

Calculator: y1= ln(x)Domain: x > 0 or x (0,∞)Range: y or y (-∞,∞)Zeros: (1,0) or x = 1X-Intercept: (1,0)Y-Intercept: none

Logarithmic Function (Natural)

( ) lnf x x

Symmetry: NoneMax: NoneMin: None Increasing: x (0,∞)Decreasing: NeverVertical Asymptotes: x = 0Horizontal Asymptotes: None

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)

( ) logbf x x

Definition of LogarithmIf and are positive numbers, 1 and 0,

then if and only if logybb x x

x b

y

b x

Exponential Form Logarithmic Form

(base)(exp) = (product) (exp) = log(base)(product)

EX: 5y = 25 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

Example 1….Change from Log notation to Exponential notation or Exponential notation to Log notation.

Logarithmic Form Exponential Form

39 2Log 23 9

8 3xLog 3 8x

811

29Log

1281 9

5 25Log x 5 25x

(___ is the exp. that takes a base of __ to a __.)

(___ is the exp. that takes a base of __ to an __.)

2 3 9

3 x 8

Example 2….Change from Log notation to Exponential notation or Exponential notation to Log notation.Logarithmic Form Exponential Form

381 4Log 43 81

82

34Log

238 4

35Log x 3 5x

(__ is the exp. that takes a base of __ to an ___.)4 3 81

Equivalent Equations x = 3y and y = log3x

Only One Graph Is Visible.

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.

Inverse Equations

Two graphs are visible that are reflected over the y = x line.

y = 2x and y = log2x

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

2 3xy

2 3yx

3 2yx

2log ( 3)y x

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.

Example 4Determine the inverse of the following. Your final answer should be in the equation box

below.

A. y = 4x-2.

x = 4y-2.

x+2 = 4y

y=log(x+2)

B. log2x+3

x = log2y+3

x-3 = log2y

y=2x-3

Example 4Determine the inverse of the following.

C. y = 6x-5

x = 6y-5

log6x= y-5

B. y = log6(x+2)

x = log6(y+2)

6x = y+2

y=6x - 2 log6x+5=yOr

y = log6x+5

Example 5….Evaluate the expression.

4log 64

3. x

4log 64 x

4 64x Re-write in exponential form.

Set equal to x

3Since 4 64, Solve for x.

Example 5…2Solve the equation log 3 for .x x

2log 3x 32 x

8x

Definition of Logarithms

Example 6…2Solve the equation log 32 3 for .x x

2log 32 3x

3 52 2x

53

x

Definition of Logarithms32 32x

3 5x

Property of Equality

Equate the Exponents

Solve for x

Cont…2Solve the equation log 32 3 for .x x

25

3log 32 3( )

2log 32 5

32 32

52 32

The solution checks!

Example 7…1Solve the equation log 2 for .9

x x

2 2 1(3 ) x

3x

1log 29

x

2 19

x

2 23 x

Definition of Logarithms

Power of Power

Equate the Exponents

Check your solution!

Example 7…1Solve the equation log 2 for .9

x x

2

1 13 9

31log 29

2 139

1 19 9

The solution checks!

Forms: Logarithmic FunctionsParent Function:

Common Log:

Natural Log:

Standard Form:

Transfm Form:

10log logy x y x

lnlogey x y x

xy blog

khxay b log

cbxay 10log

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