Chabot Mathematics
§9.3b
Base 10 & e Logs
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt
Review § 9.3
MTH 55
 Any QUESTIONS About
• §9.3 → Introduction to Logarithms
 Any QUESTIONS About HomeWork
• §9.3 → HW-44
Chabot College Mathematics
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt
Common Logarithms


The logarithm with base 10 is called
the common logarithm and is denoted
by omitting the base: logx = log10x. So
y = logx if and only if x = 10y
Applying the basic properties of logs
1. log(10) = 1
2. log(1) = 0
3. log(10x) = x
4. 10logx = x
Chabot College Mathematics
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt
Common Log Convention
 By this Mathematics CONVENTION the
abbreviation log, with no base written, is
understood to mean logarithm base 10,
or a common logarithm. Thus,
log21 = log1021
 On most calculators, the key for
LOG
common logarithms is marked
Chabot College Mathematics
4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt
Example  Calc Common Log

Use a calculator to approximate each
common logarithm. Round to the
nearest thousandth if necessary.
a. log(456)
b. log(0.00257)

Solution by Calculator LOG key
•
log(456) ≈ 2.659 → 102.659 = 456
•
log(0.00257) ≈ −2.5901 → 10−2.5901 =
0.00257
Chabot College Mathematics
5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt
Example  Calc Common Log
 Use a scientific calculator to
approximate each number to 4 decimals
log130
b)
log(0.35)
a) log 2,356

Use a scientific calculator to find
a) log 2,356  3.3722.
log130
b)
 4.6365.
b)
log(0.35)
Chabot College Mathematics
6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt
Example  Sound Intensity
 This function is sometimes
 I 
used to calculate
d  10log  
sound intensity
 I0 
 Where
• d ≡ the intensity in decibels,
• I ≡ the intensity watts per unit of area
• I0 ≡ the faintest audible sound to the
average human ear, which is 10−12 watts
per square meter (1x10−12 W/m2).
Chabot College Mathematics
7
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt
Example  Sound Intensity
 Use the Sound Intensity Equation
(a.k.a. the “dBA” Eqn) to find the
intensity level of sounds at a decibel
level of 75 dB?
 Solution: We need
to isolate the
intensity, I, in
the dBA eqn
Chabot College Mathematics
8
 I 
d  10log   ,
 I0 
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt
Example  Sound Intensity
 Solution (cont.) in the dBA eqn
substitute 75 for d and 10−12 for I0 and
then solve for I
 I 
75  10  log 
12 
 10

 I  1012  I  1012  107.5
7.5  log 
12
12 
10
 10

I
4.5
7.5
10
I
 10
10
12
Chabot College Mathematics
9
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt
Example  Sound Intensity
 Thus the Sound
Intensity at 75 dB is
10−4.5 W/m2 =
10−9/2 W/m2
 Using a Scientific
calculator and find
that I = 3.162x10−5
W/m2
= 31.6 µW/m2
Chabot College Mathematics
10
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt
Example  Sound Intensity
 Check
If the sound intensity is 10−4.5 W/m2 ,
verify that the decibel reading is 75.
 104.5 
d  10log 

 1012 


7.5
d  10log10
d  10  7.5
Chabot College Mathematics
11
d  75

Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt
Graph log by Translation
 Sketch the graph of y = 2 − log(x − 2)
 Soln: Graph f(x) = logx and shift Rt 2 units
f x   log x 
Chabot College Mathematics
12
f x   log x  2 
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt
Graph log by Translation
 Reflect in x-axis
y   log x  2 
Chabot College Mathematics
13
 Shift UP 2 units
y  2  log x  2 
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt
Example  Total Recall
 The function P = 95 – 99∙logx models
the percent, P, of students who recall
the important features of a classroom
lecture over time, where x is the number
of days that have elapsed since the
lecture was given.
 What percent of the students recall the
important features of a lecture 8 days
after it was given?
Chabot College Mathematics
14
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt
Example  Total Recall
 Solution:
Evaluate P = 95 – 99logx when x = 8.
P = 95 – 99log(8)
P = 95 – 99(0.903) [using a calculator]
P = 95 – 89
P=6
 Thus about 6% of the students
remember the important features of a
lecture 8 days after it is given
Chabot College Mathematics
15
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt
Natural Logarithms
 Logarithms to the base “e” are called
natural logarithms, or Napierian
logarithms, in honor of John Napier,
who first “discovered” logarithms.
 The abbreviation “ln” is generally used
with natural logarithms. Thus,
ln 21 = loge 21.
 On most calculators, the key for natural
logarithms is marked LN
Chabot College Mathematics
16
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt
Natural Logarithms


The logarithm with base e is called the
natural logarithm and is denoted by ln
x. That is, ln x = loge x. So
y = lnx if and only if x = ey
Applying the basic properties of logs
1. ln(e) = 1
2. ln(1) = 0
3. ln(ex) = x
4. elnx = x
Chabot College Mathematics
17
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt
Example  Evaluate ln
 Evaluate each expression
1
4
b. ln 2.5
a. ln e
e
 Solution
4
a. ln e  4
c. ln 3
1
2.5
b. ln 2.5  ln e  2.5
e
c. ln 3  1.0986123
Chabot College Mathematics
18
(Use a calculator.)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt
Example  Compound Interest
 In a Bank Account that Compounds
CONTINUOUSLY the relationship
between the $-Principal, P, deposited,
the Interest rate, r, the Compounding
time-period, t, and the $-Amount, A, in
the Account:
1 A
t  ln
r P
Chabot College Mathematics
19
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt
Example  Compound Interest
 If an account pays 8% annual interest,
compounded continuously, how long will
it take a deposit of $25,000 to produce
an account balance of $100,000?
 Familiarize
In the Compounding Eqn replace P with
25,000, r with 0.08, A with $100,000,
and then simplify.
Chabot College Mathematics
20
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt
Example  Compound Interest
 Solution
1
100, 000
t
ln
0.08 25, 000
1
t
ln 4
0.08
t  17.33
Substitute.
Divide.
Approximate using a calculator.
 State Answer
The account balance will reach
$100,000 in about 17.33 years.
Chabot College Mathematics
21
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt
Example  Compound Interest
 Check:
1
A
17.33 
ln
0.08 25, 000
A
1.3864  ln
25, 000
1.3864  ln A  ln 25, 000
1.3864  ln 25, 000  ln A
11.513  ln A
e11.513  A
100, 007.5  A
 Because 17.33 was not the exact time,
$100,007.45 is reasonable for the Chk
Chabot College Mathematics
22
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt
WhiteBoard Work
 Problems From §9.3 Exercise Set
• 52, 58, 64, 70,
72, 90
 Loud Noise
Safe Exposure
Time
Chabot College Mathematics
23
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt
All Done for Today
“e”
to Several
Digits
Chabot College Mathematics
24
e=
2.7182818284590452353602874713526624
97757247093699959574966967627724076
63035354759457138217852516642742746
63919320030599218174135966290435729
00334295260595630738132328627943490
76323382988075319525101901157383418
79307021540891499348841675092447614
60668082264800168477411853742345442
43710753907774499206955170276183860
62613313845830007520449338265602976
06737113200709328709127443747047230
69697720931014169283681902551510865
746377211125238978442505695369677078
54499699679468644549059879316368892
30098793127736178215424999229576351
48220826989519366803318252886939849
64651058209392398294887933203625094
43117301238197068416140397019837679
32068328237646480429531180232878250
9819455815301756717361332069811250
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
25
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt