Transcript Intro to logs - FHS PAP Algebra 2

Logarithms
and
Logarithmic Functions
Unit 6.6
Warm up…
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.
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.
Warm up…
1.
2.
y
1
10
 10
x
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)
f ( x )  log 10 x
b  1; b  10
Calculator: y1= log(x)
Domain: x > 0 or x  (0,∞)
Range: y 
or y  (-∞,∞)
Zeros: (1,0) or x = 1
X-Intercept: (1,0)
Y-Intercept: none
Logarithmic Function
(Common)
f ( x )  log 10 x
b  1; b  10
Symmetry: None
Max: None
Min: None
Increasing: x  (0,∞)
Decreasing: Never
Vertical Asymptotes: x = 0
Horizontal Asymptotes: None
Logarithmic Function
(Natural)
f ( x )  ln x
Calculator: y1= ln(x)
Domain: x > 0 or x  (0,∞)
Range: y 
or y  (-∞,∞)
Zeros: (1,0) or x = 1
X-Intercept: (1,0)
Y-Intercept: none
Logarithmic Function
(Natural)
f ( x )  ln x
Symmetry: None
Max: None
Min: None
Increasing: x  (0,∞)
Decreasing: Never
Vertical Asymptotes: x = 0
Horizontal Asymptotes: None
Logarithmic Functions
f ( x )  log b x
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)
Definition of Logarithm
If b and x are positive num bers, b  1 and x  0,
then b  x if and only if log b x  y
y
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
Log 3 9  2
3 9
2
(___
2 is the exp. that takes a base of __
3 to a __.)
9
Log x 8  3
(___
3 is the exp. that takes a base of __
x to an __.)
8
L og 81 9 
1
x 8
3
1
81  9
2
2
Log 5 25  x
5  25
x
ExamplE 2….
Change from Log notation to Exponential
notation or Exponential notation to Log
notation.
Logarithmic Form
Log 3 81  4
Exponential Form
3  81
4
(__
4 is the exp. that takes a base of __
3 to an ___.)
81
L og 8 4 
2
3
Log 3 5  x
2
83  4
3 5
x
Equivalent Equations
x = 3y and y = log3x
Only One Graph Is Visible.
Inverse Equations
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.
y = 2x and y = log2x
Two graphs are visible that are reflected over the y = x line.
Finding the inverse algebraically
Example 3: Rewrite the inverse of the exponential
function y = 2x + 3 as a logarithmic function.
Beginning Equation
y  2 3
Replace x with y and y with x.
x  2 3
Isolate the term containing y.
x3 2
Rewrite the exponential function
as a logarithmic function
y  log 2 ( x  3)
x
y
y
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 4
Determine the inverse of the following.
Your final answer should be in the equation box below.
A. y = 4x-2.
B. log2x+3
x = 4y-2.
x = log2y+3
x+2 = 4y
x-3 = log2y
y=log(x+2)
y=2x-3
Example 4
Determine the inverse of the following.
C. y = 6x-5
B. y = log6(x+2)
x = 6y-5
x = log6(y+2)
log6x= y-5
log6x+5=y
Or
y = log6x+5
6x = y+2
y=6x - 2
ExamplE 5….
Evaluate the expression.
lo g 4 6 4  x
4  64
x
Since 4  64,
3
x  3.
lo g 4 6 4
Set equal to x
Re-write in exponential form.
Solve for x.
Example 5…
S o lve th e eq u atio n lo g 2 x  3 fo r x .
lo g 2 x  3
2  x
3
x 8
Definition of Logarithms
Example 6…
S o lve th e eq u atio n lo g 2 3 2  3 x fo r x .
lo g 2 3 2  3 x
2
3x
 32
Definition of Logarithms
3x
2
Property of Equality
2
3x  5
x
5
3
5
Equate the Exponents
Solve for x
Cont…
S o lve th e eq u atio n lo g 2 3 2  3 x fo r x .
5
log 2 32  3( )
3
lo g 2 3 2  5
2  32
5
32  32
The solution checks!
ExamplE 7…
S olve the equation log x
1
log x
x
x
x
  2 for x .
9
 2
9
2

1
Definition of Logarithms
9
2
1
 (3 )
2
2
3
1
2
x 3
Power of Power
Equate the Exponents
Check your solution!
ExamplE 7…
S olve the equation log x
log 3
3
1
1
  2 for x .
9
 2
9
2

1
9
 1  1
 2  
3  9
1 1
  
9 9
The solution checks!
Forms: Logarithmic
Functions
Parent Function: y  log
b
x 
Common Log:
y  log 10 x  y  log x
Natural Log:
y  log e x  y  ln x
x  h   k
Standard Form:
y  a  log
Transfm Form:
y  a  log 10  x  b   c
b
Absent Students-Notes 6.6
Attach this note page
into your notebook
Complete all
examples and warmups
Be sure to
understand the
difference between
finding the inverse,
evaluating, solving and
rewriting logs and
exponentials