MAT 150 Module 9 – Logarithmic
Functions
Lesson 1 –
Logarithmic functions
and their properties
Definition of a Logarithm
The logarithmic function
is the inverse of the
exponential function.
Instead of knowing the
power, we are trying to
find the power.
Exponential function
Y = ax
Raise the base, a, to a
power, x, to get the
output, y.
Logarithmic function
Y = logax
We know the output
and we want to find the
power, x.
Definition of a Logartithm
Logax = ?
is really asking
?
a =
x
Logarithmic form and exponential
form
Exponential
Form
ax = y
Logarithmic
Form
logay = x
Graphing
Example 1
Rewrite in exponential form:
logpy = B
Example 1
Rewrite in exponential form:
logpy = B
P is the base
B is the
exponent
pB = y
Y is the
output
Example 2
Rewrite in logarithmic form:
w = r2
Example 2
Rewrite in logarithmic form:
w = r2
r is the base
2 is the
exponent
logrw = 2
w is the
output
Example 3
Evaluate without a calculator:
1. Log2512 =
2.
3. Log 10,000 =
4. Log77 =
5. Log2 1=
6. Log7 -5 =
Example 3 - Solution
Evaluate without a calculator:
1. Log2512 = 9
2 is the base and
512 is the output.
2.
5 is the base and
1/25 is the output.
Think: 2? = 512
29 = 512
-2
Think: 5? = 1/25
5-2 = 1/25
Example 3
Evaluate without a calculator:
3. Log 10,000 =4
10 is the base and
10,000 is the
output.
Think: 10? =
10,000
104 = 10,000
Think: 7? = 7
71 = 7
4.Log77 = 1.
7 is the base and 7
is the output.
Example 3
Evaluate without a calculator:
5. Log2 1= 0
2 is the base and 1
is the output.
Think: 2? = 1
20 = 1
6. Log7 -5 = Not possible
7 is the base and
-5 is the output.
Think: 7? = -5
7x cannot
equal -5 for any
power
The Natural Logarithm
The logarithmic function
with base e is called the
Natural Logarithm and is
written ln(x).
When you see ln(x),
think of it as Loge x.
Evaluating Logs with a calculator
Most scientific
calculators have two
buttons
To calculate any other
bases, use the change
of base formula
log
Base 10
ln
Base e
Example 4
Evaluate with a calculator. Round to four
decimal places.
1. Log 75 =
2. Ln 3.175 =
3. Log215 =
4. log579 =
Example 4 - Solution
Properties of Logarithms
Logarithms have some interesting properties
that allow us to simplify and rewrite them.
Before calculators existed, these properties
were used as a shortcut to do complicated
multiplication and division problems.
Basic Properties of Logarithms
. loga(a) = 1
since a1 = a for any a
loga(1) = 0
since a0 = 1 for any a
loga(ax) = x
since ax = ax for any a
and any x
since ax = ax for any a
and any x
Basic Properties - Examples
1. Log66 = 1 since 61 = 6
2. Log201 = 0 since 200 = 1
3. Log552= 2 since 52 = 52.
4.
= 26 since 326 = 326
The Addition and Subtraction
Properties
Recall from what we know about exponents
that :
bM • bN = bM+N
The Addition property of logarithms is:
logb(MN) = logbM + logbN
The Addition and Subtraction
Properties
Recall from what we know about exponents
that :
The subtraction property of logarithms is:
Addition and Subtraction Properties
– Example 1
Rewrite as the sum or difference of two
logarithms.
a. Log5(25x) =
b. Log4
=
Addition and Subtraction Properties
– Example 1 Solution
Rewrite as the sum or difference of two
logarithms.
Log5(25x) = Log5(25) + Log5(x)
Using the addition property.
Then we know Log5(25) = 2 since 52 = 25.
Log5(25x) = 2 + Log5(x)
Addition and Subtraction Properties
– Example 1 Solution
Rewrite as the sum or difference of two
logarithms.
Log4
= Log4P - Log4Q
using the subtraction property.
The Power Property
This property tells us that logarithms turn
exponents into coefficients:
n
logax
= n*logax
The Power Property – Example 2
Rewrite the following using the power
property:
a. Log4(x5) =
b. Ln(c7) =
c. 8log6p =
The Power Property – Example 2
Solutions
Rewrite the following using the power
property:
a. Log4(x5) = 5Log4(x)
b.
Ln(c7)
= 7Ln(c)
c. 8log6p = log6
p8
Bring down the exponent of 5
and make it a coefficient.
Bring down the exponent of 5
and make it a coefficient.
Using the property in reverse,
bring up the coefficient of 8
and make it an exponent.
Using the properties together
We can use any or all of the logarithms
properties together to combine logarithms or
rewrite them as a sum/difference of
logarithms.
Example 3
Use the properties of logarithms together to
rewrite the expression.
a. Log3(x2y3) =
b. Log7
Example 3
Use the properties of logarithms together to
rewrite the expression.
a. Log3(x2y3)
First, use the addition
property since x2y3 is
multiplied.
Then use the power
property to bring down the
powers and make them
coefficients.
Log3(x2y3) = Log3(x2) + Log3(y3)
Log3(x2) + Log3(y3)
= 2Log3(x) + 3Log3(y)
Example 3
Use the properties of logarithms together to
rewrite the expression.
b. Log7
First, use the subtraction
property since x5√m is divided
by k.
Next, use the additionproperty
Log7[(x5√m)/k] = Log7(x5√m) - Log7(k)
since x5√m is multiplied.
= Log7(x5) + Log7(√m) - Log7(k)
Then use the power property to
bring down the powers and
make them coefficients.
= 5Log7(x) + ½Log7(m) - Log7(k)
Example 4
Use the properties of logarithms together to
rewrite the expression.
a. 2 Log2 w + 5 Log2 p =
b. 5 ln (x) + 2 ln (y) - 3 ln (z)=
Example 4 - Solution
Use the properties of logarithms together to
rewrite the expression.
a. 2 Log2 w + 5 Log2 p =
First, use the power property
to bring up the coefficients
and make them powers.
Next use the addition property
to combine the two logs that
are added into one where w
and p are multiplied.
2 Log2 w + 5 Log2 p = Log2 w2 + Log2 p5
Log2 w2 + Log2 p5 = Log2 (w2 p5)
Example 4
Use the properties of logarithms together to
rewrite the expression.
a. 5 ln (x) + 2 ln (y) - 3 ln (z)=
First, use the power property
to bring up the coefficients
and make them powers.
Next use the addition and
subtraction properties to
combine the three logs into one
where x and y are multiplied
and z is divided.
= Ln(x5) + Ln(y2) – Ln(z3)