Download Properties of Logarithms

Properties of
Logarithms
They’re in Section 3.4a
Proof of a Prop ‘o Logs
Let
x  logb R
In exponential form:
and
y  logb S
b R
x
b S
y
Let’s start with the product of R and S:
RS  b  b
x y
RS  b
logb  RS   x  y  logb R  logb S
x
y
A Prop ‘o Logs!!!
Properties of Logarithms
Let b, R, and S be positive real numbers with b = 1, and c
any real number.
• Product Rule:
logb  RS   logb R  logb S
• Quotient Rule:
R
log b  log b R  log b S
S
• Power Rule:
log b R  c log b R
c
Guided Practice
Assuming x and y are positive, use properties of logarithms
to write the given expression as a sum of logarithms or
multiples of logarithms.
log  8xy
4
  log8  log x  log y
4
 log 2  log x  log y
3
4
 3log 2  log x  4 log y
Guided Practice
Assuming x is positive, use properties of logarithms to write
the given expression as a sum or difference of logarithms
or multiples of logarithms.
x  5

x 5
 ln
ln
x
x
2
2
12
 ln  x  5  ln x
2
12
1
2
 ln  x  5   ln x
2
Guided Practice
Assuming x and y are positive, use properties of logarithms
to write the given expression as a single logarithm.
5ln x  2ln  xy   ln x  ln  xy 
5
2
x
x
 ln x  ln  x y   ln 2 2  ln 2
x y
y
5
5
2
2
3
Of the eight relationships suggested here, four are true and four
are false (using values of x within the domains of both sides of
the equations). Thinking about the properties of logarithms,
make a prediction about the truth of each statement. Then test
each with some specific numerical values for x. Finally, compare
the graphs of the two sides of the equation.
1. ln x  2  ln x  ln 2
2. log3 7 x  7log3 x


3. log 2  5x   log 2 5  log 2 x
x log x
5. log 
4 log 4
7.
log5 x   log5 x  log5 x 
2
 
4.
x
ln  ln x  ln 5
5
6.
log 4 x 3  3log 4 x
8.
log 4x  log 4  log x
These four statements are TRUE!!!
A few more problems…
Assuming x and y are positive, use properties of logarithms to
write the expression as a sum or difference of logarithms or
multiples of logarithms.
log1000x  log1000  log x  3  4 log x
4
3
ln
5
x
y
2
4
 ln x  ln y
13
25
1
2
 ln x  ln y
3
5
And still a few more problems…
Assuming x, y, and z are positive, use properties of logarithms to
write the expression as a single logarithm.
4
1
 y 
4
4 log y  log z  log y  log z  log 

2
 z
3ln  x yz
  2 ln  yz 
 ln  x y z   ln  y z   ln  x y z 
3
2
2
9
3 6
2 4
9
5 10
Whiteboard…
Write as a single logarithmic expression:
 log 100 log3x  5log x
 2log 3x  5log x
2
 log  3x   log  x
2
2

2 5
 log 9 x  log x
2
 9x 
 9
 log  10   log  8 
x 
x 
2
10
Let’s do an exploration…
How do we evaluate
4 7
y
log 4 7 ?
Set equal to y:
y  log 4 7
First, switch to exponential form.
ln 4  ln 7 Apply ln to both sides.
y ln 4  ln 7 Use the power rule. We just proved
the C.O.B.!!!
ln 7
y
 1.404 Divide by ln4.
ln 4
y
Change-of-Base Formula for
Logarithms
For positive real numbers a, b, and x with a = 1
and b = 1,
a
b
a
Because of our calculators, the two most common forms:
log x
log x 
log b
log x ln x
logb x 

log b ln b
Guided Practice
Evaluate each of the following.
1.
2.
3.
ln16
log3 16 
 2.524
ln 3
1
log10

 1.285
log 6 10 
log 6 log 6
ln 2
ln 2
 1

log1 2 2 
ln 1 2   ln 2
Guided Practice
Write the given expression using only natural logarithms.
ln 3 x
1. log5 3x 
ln 5
2.
log7  2x  y  
ln  2 x  y 
ln 7
Guided Practice
Write the given expression using only common logarithms.
log s
1. log 4 s 
log 4
2.
log1 4  a  2b  
log  a  2b 
log 1 4 
log  a  2b 

log 4
Graphs of Logarithmic Functions with
Base b
Rewrite the given function using the change-of-base formula.
ln x
1

ln x
g  x   logb x 
ln b ln b
 Every logarithmic function is a constant multiple of the
natural logarithmic function!!!
If b > 1, the graph of g(x) is a vertical stretch or shrink of the
graph of the natural log function by a factor of 1/(ln b).
If 0 < b < 1, a reflection across the x-axis is required as well.
More Guided Practice
Describe how to transform the graph of the natural logarithm
function into the graph of the given function. Sketch the graph
by hand and support your answer with a grapher.
1
1. g  x   log x 
ln x
5
ln 5
1
Vertical shrink by a factor
 0.621
of approximately 0.621.
ln 5
How does the graph look???
More Guided Practice
Describe how to transform the graph of the natural logarithm
function into the graph of the given function. Sketch the graph
by hand and support your answer with a grapher.
1
ln x


ln
x

2. h  x   log
x
14
ln 4
ln1 4
1
Reflect across x-axis, Vertical
 0.721
shrink by a factor of 0.721
ln 4
How does the graph look???