1.
2.
3.
4.
Objectives:
To apply the definition
of logarithm
To find the inverses of
logarithmic and
exponential functions
To graph logarithmic
functions as inverses
of exponentials
To solve simple
logarithmic equations
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•
•
•
•
•
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Assignment:
P. 236: 1-8 (Some)
P. 236: 9-16 (Some)
P. 236: 17-22 (Some)
P. 236: 27-30
P. 236: 31-38 (Some)
P. 237: 47, 55
P. 237: 79-86
Homework Supplement
Find the value of x.
1. 2x = 8
2. 2x = 9
Find the value of x.
1. 2x = 8
2. 2x = 9
When you ask “2 to what power
is 9?” you can’t come up
with an answer.
To solve for x, we use a
logarithm, which is the
inverse of an exponential.
Use your graphing
utility to graph
each of the
following on the
same screen.
1. y = 10x
2. y = log x
3. y = x
Use your graphing
utility to graph
each of the
following on the
same screen.
1. y = ex
2. y = ln x
3. y = x
As the previous exercises
showed, exponential
and logarithmic
functions are inverses.
So to graph a
logarithmic function:
1. Graph the exponential
2. Switch the coordinates
of some key points
(But first, maybe we should discuss
the definition of logarithm.)
You will be able to apply the
definition of logarithm
Logarithm with Base b
For y > 0, b > 0, and b ≠ 1, the logarithm with
base b of y is denoted as logb y such that
logb y = x if and only if bx = y
The base of the logarithm is the base of the exponential equation
Logarithm with Base b
For y > 0, b > 0, and b ≠ 1, the logarithm with
base b of y is denoted as logb y such that
logb y = x if and only if bx = y
The answer to the logarithm is the exponent of the exponential
Logarithm with Base b
For y > 0, b > 0, and b ≠ 1, the logarithm with
base b of y is denoted as logb y such that
logb y = x if and only if bx = y
The general logarithm function is an exponent-producing
function. The log base b is the exponent you have to put on
b to get y.
Logarithm with Base b
For y > 0, b > 0, and b ≠ 1, the logarithm with
base b of y is denoted as logb y such that
logb y = x if and only if bx = y
Logarithmic
Form
Exponential
Form
Since
logb y = x if and only if bx = y
we can use the definition to rewrite a
logarithmic equation in exponential form.
log5 125  3
53  125
Rewrite each equation in exponential form.
1. log2 32 = 5
2. log10 1 = 0
3. log9 9 = 1
4. log1/5 25 = -2
5. log7 73 = 3
Let b be a positive real number with b  1.
1. Since b0 = 1,
logb 1 = 0
2. Since b1 = b,
logb b = 1
3. Since ba = ba,
logb ba = a
Rewrite each equation in exponential form.
1. log3 81 = 4
2. log7 7 = 1
3. log14 1 = 0
4. log1/2 32 = -5
5. log8 87 = 7
Since logarithms are essentially exponents, to
evaluate a log like log6 216, you could:
1. Ask “6 to what power
equals 216?”
6  216
3
2. Rewrite in exponential
form:
log 6 216  x
6  216
3
6  216
x
Evaluate the logarithm.
1. log3 81
2. log1/4 256
3. log10 0.001
4. log64 2
Special logarithms are logs with their own
calculator buttons: base 10 or base e.
Common Log
Natural Log
log10 a  log a
log e a  ln a
log y  x  10  y
ln y  x  e  y
x
x
Use a calculator to evaluate the following.
1. log 0.85
2. ln 22
Evaluate each logarithm.
1. log2 32
2. log27 3
3. log 12
4. ln 0.75
Logarithmic functions are inverses of
exponential functions.
Inverse Properties
log b b  x
x
b
logb x
x
Simplify the expression.
1. eln 9
2. log3 27x
Simplify the expression.
1. 8log8 x
2. log7 7-3x
3. log2 64x
4. eln 20
To write the inverse of a logarithmic function:
1. Switch x and y
2. Write in exponential form
To write the inverse of an exponential function:
1. Switch x and y
2. Write in logarithmic form
Find the inverse function.
Find the inverse function.
Notice that every logarithmic equation has both
an equivalent exponential equation and an
inverse, which is also exponential.
y  logb x
Equivalent
Inverse
xb
y b
y
x
You will be able to graph a
logarithmic function
Perhaps the easiest way to graph a logarithmic
function is to first plot its inverse. This will be
an exponential function. Then switch the xand y-coordinates of some key points.
1. Plot the inverse,
which is an
exponential function.
Graph the function. State the domain and
range.
1. y = log2 x
2. y = log1/3 x
Graph of y = logb x, b > 1
Domain: (0, ∞)
Range: (−∞, ∞)
Vertical asymptote: x = 0
x-intercept: 1
Increasing
Continuous
One-to-one
For each function, state the domain, x-intercept,
and vertical asymptote.
You will be able to solve simple logarithmic equations
Horizontal Line Test
One-to-One
Every input has 1 output
Every output has 1 input
Enables us to solve simple
logarithmic equations
when the bases are equal.
If log5 x = log5 16, what is the value of x?
Use the One-to-One Property to find x.
1. log(4  3x)  log( x  2)
2. log 3 ( x 2  4)  log 3 29
1.
2.
3.
4.
Objectives:
To apply the definition
of logarithm
To find the inverses of
logarithmic and
exponential functions
To graph logarithmic
functions as inverses
of exponentials
To solve simple
logarithmic equations
•
•
•
•
•
•
•
•
Assignment:
P. 236: 1-8 (Some)
P. 236: 9-16 (Some)
P. 236: 17-22 (Some)
P. 236: 27-30
P. 236: 31-38 (Some)
P. 237: 47, 55
P. 237: 79-86
Homework Supplement