Logarithmic Functions
Logarithm = Exponent
Very simply, a logarithm is an exponent of
ten that will produce the desired number.
Y = Log 100 means what is the exponent
of 10 which will produce 100.
Y = log .001 means what is the
exponent of 10 which will produce
.001.
Note: when no base is indicated, the
base is 10.
Y=logb a can be read “ y is the exponent of
base b to produce a.”
3= log2 8 is read “3 is the exponent of 2
to produce 8”
2 = log4 16
1
2  log 2
4
 42 = 16

2
2
1

4
Write the equivalent exponential equation
and solve for y.
Logarithmic
Equation
Equivalent
Exponential
Equation
Solution
16 = 2y
16 = 24  y = 4
1
2
1 -1
= 2  y = –1
2
y = log416
16 = 4y
16 = 42  y = 2
y = log51
1=5y
1 = 50  y = 0
y = log216
1
y = log2(
2
)
=
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2y
4
Definition of a Logarithm
• A logarithm, or log, is defined in terms of
an exponent.
• If b x=a, then logb a =x
If 5 2=25 then log5 25=2
Log5 25=2 is read “log base 5 of 25 is 2.”
– You might say the log is the exponent we
apply to 5 to make 25
Log of a Product
• The log of a product is the sum of the
logs of the factors
logbxy = logbx + logby
Log2512 = log2(8·64)
= log28 + log264
=3
+ 5
= 8
Log of a Quotient
The log of a quotient is the difference
of the logs of the factors.
x
logb  logb x  logb y
y
125 

Ex. log5 
  log 5 125  log 5 25
 25 
 3 2
 1
Log of a Power
The log of a power is the product of the
exponent and the log of the base.
logbxn = nlogbx
Ex: log 32 = 2log3
Given:
loga 2  .3456
loga 5  .4985
loga 3  .4234
log a 7  .6002
Use the properties of logs to simplify
the following:
log a 15
log a 42
5
log a
3
2
log a
7
log a 7
4
log a
 2  5
3
For x  0 and 0  a  1,
y = loga x if and only if x = a y.
The function given by f (x) = loga x is called
the logarithmic function with base a.
Every logarithmic equation has an equivalent
exponential form:
y = loga x is equivalent to x = a y
A logarithmic function is the inverse function
of an exponential function.
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10
The logarithm function f (x) = log10 x is
called the common logarithm function.
Function
Value
Keystrokes
log10 100
LOG 100
log10(
2
5
log10 5
log10 –4
)
LOG –4
2
ENTER
LOG ( 2 / 5 )
LOG 5
Display
ENTER
ENTER
ENTER
– .3979400
0.6989700
ERROR
REMEMBER: no power of 10 gives a
negative number
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11
Properties of Logarithms
1. loga 1 = 0 since a0 = 1.
2. loga a = 1 since a1 = a.
3. If loga x = loga y, then x = y. one-to-one
property
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12
INVERSE PROPERTIES:
The logarithm with base a of a raised to a
power equals that power:
x
a
log a  x
a raised to the logarithm with base a of a
number equals that number
a
loga x
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x
13
One way to Graph f (x) = log2 x
Since the logarithm function is the inverse of
the exponential function of the same base, its
graph is the reflection of the exponential
x
y
y
=
2
x the line y = x.
y=x
function
x
2in
1
–2
horizontal
4
y = log2 x
1
asymptote y = 0
–1
2
x-intercept
0
1
x
(1, 0)
1
2
2
4
vertical asymptote
3
8
x=0
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14
Graphing Log Functions
One way to Graph f (x) = log2 x
Since the logarithm function is the inverse of
the exponential function of the same base, its
graph is the reflection of the exponential
x
y
y
=
2
x the line y = x.
y=x
function
x
2in
1
–2
horizontal
4
y = log2 x
1
asymptote y = 0
–1
2
x-intercept
0
1
x
(1, 0)
1
2
2
4
vertical asymptote
3
8
x=0
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16
Graph of the common logarithm function f(x) =
log10 x.
1
1
x
1
2
4
10
100
f(x) = log10 –2
x
10
–1
0
0.301 0.602 1
by
f(x)calculator
= log10 x
y
x
5
(0, 1) x-intercept
x=0
vertical
asymptote
–5
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17
The graphs of logarithmic functions are
similar for different values of a.
f(x) = loga x (a  1)
Graph of f(x) = loga x (a
(0, )

1. 1)
2.
domain(,)
3.
x-intercept (1, 0)
range
4. vertical 
x  0 as x  0 f ( x)  
asymptote
5.
6.
increasing
7.
one-tocontinuous
8.
onereflection of y = a x in y
=x
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y-axis
vertical
asympto
te
yy = a x
y=
x
y = log2
x
domain
x
x-intercept
(1, 0)
range
18
Using the graphing calculator to
graph the log functions
• For other than base 10, use the following
formula:
log10 x
log a x 
log10 a
To graph the function
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19
Graph the following functions:
f ( x)  log 2 x
f ( x)  log 3 x
f ( x)  log 4 x
f ( x)  log 5 x
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20
y  log 2 x
y  log 4 x
y  log3 x
y  log5 x
y
The function defined
by f(x) = loge x = ln x
5
(x  0, e .718281)
is called the
natural
–5
logarithm
function. y = ln x is equivalent to e y
=x
Use a calculator to evaluate: ln 3, ln –2,
Keystrokes
ln Function
100
Value
ln 3
LN 3 ENTER
ln –2
LN –2 ENTER
ln 100
LN 100 ENTER
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y = ln x
x
Display
1.0986122
ERROR
4.6051701
22
Properties of Natural Logarithms
1. ln 1 = 0 since e0 = 1.
2. ln e = 1 since e1 = e.
3. ln ex = x and eln x = x
inverse property
4. If ln x = ln y, then x = y. one-to-one property
Examples: Simplify each expression.
 
1
ln  2   ln e 2  2
e 
inverse property
e ln 20 20
inverse property
3 ln e  3(1)  3
property 2
ln 1  0  0
property 1
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23
.
.