Section 5-5 Logarithmic
Functions pg.191
Objective : To define and apply
logarithms
Now we will explore logarithms,
numbers used to measure the severity
of earthquakes, the loudness of sounds,
and brightness of suns.
We define the common logarithm of an
integral power of 10 to be its exponent
as shown below.
Number
10o
101
102
103
104
105
Common
logarithms
0
1
2
3
4
…….
10k
k
We denote the common logarithm of x
by log x for example…
• Log 103 = 3, Log 104 = 4, and Log 10k = k
• In general the common logarithm of any
positive real number x is defined to be the
exponent you get when you write x as a
power of 10
log x = u and only if 10u = x
for example
log 6.3 = 0.8 because 6.3 = 100.8
You can verify this by evaluating 100.8 on a
calculator.
You can also use a scientific calculator to find
common logarithms by pressing the log key
and then enter the number whose common
logarithm you wish to find.
Sound
I
Decibel
Level
Barely Audible
Whisper
Io
0
10 Io
10
Leaves in a
breeze
102 Io
Soft recorded music
104 Io
20
40
106 Io
60
108Io
80
Subway Train
1010 Io
100
Jet at Takeoff
1012 Io
120
Pain in eardrum
1014 Io
140
Two-person
conversation
Loud Stereo Set
These are the levels of
sound each source makes
measured in common
logarithmic units
Common logarithms have base 10
Logarithms to other bases are
sometimes used. The logarithm to
base b of a positive number x is
denoted by log b x is defined to be the
exponent a that you get when you
write x as a power of b . (Note b >0)
and (b≠1 )
log b x = a if and only if x = b a
Example 2
log b x = a if and only if x = b a
log 5 25 = 2 because 25 = 5 2
log 5 125 = 3 because 125 = 5 3
log 2 1 = -3 because 2 -3 = 1
8
8
log 4 64 =
log 3 243 =
log 6 216 =
log 7 343 =
Base b exponential function :
F (x) = b x
Domain all reals
Range: Positive reals
Base b logarithmic functions
f -1 (x) = log b x
Domain : Positive reals
Range: All reals
F (x) = b x
f -1 (x) = log b x
The most important logarithmic
function in advanced mathematics and
statistics has the number e as its
base. The function is called the natural
logarithm function. The natural
logarithm function of x is usually
denoted ln x although sometimes it is
written log e x. The value of ln x can be
found by using a scientific calculator
ln x = k if and only if e k = x
ln x = k if and only if e k = x
For example ln 5 = 1.6 because e1.6 ≈ 5
Example 3
Find the value of x to the nearest hundredth
a. 10 x = 75
b. e x = 75
x = log 75 = 1.8
x = ln 75 = 4.32
Find the following using a scientific calculator
log ( 50) = ____ log 250 =____
log 36 =____
ln 50 = ______ ln 250 =___
ln 36 =_____
e3.9 = _____
e^___ =249.6
e ^____ =
HOMEWORK 1-22 ALL Pgs. 194-195