Transcript 3 3 Prop Logs

Objectives:
1. To use (and prove)
properties of
logarithms to
simplify expressions
2. To use (and prove)
the change of base
formula
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Assignment:
P. 243: 9-16 (Some)
P. 243: 17-22 (Some)
P. 243: 23-38 (Some)
P. 243: 39-60 (Some)
P. 244: 61-78 (Some)
P. 245: 93, 94
HW Supplement
As difficult as it might be, try to imagine a time
before calculators and computers. In such an
ancient world (30 to 60 years ago), people
would spend long, tedious hours doing
complex computations (multiplying, mostly)
by hand, often with calloused and aching
fingers. These calculations were necessary to
save the world (sail ships, calculate the path of
celestial bodies, etc.).
(Probably wearing a kilt)
Sometime in the early 17th
century, a Scottish
mathematical hobbyist
named John Napier
(1550-1617) found a
shortcut through those
calculations involving
what he called
logarithms.
(Probably wearing a kilt)
According to something I
read once, his discovery
was the equivalent to
the invention of the
computer.
These shortcuts involved a
table of logs (not
wooden) and a few good
properties.
According to
the table of
logs, what is
log (2.65)?
According to
the table of
logs, what is
log (2.65)?
Confirm the
answer with
your
calculator.
According to
the table of
logs, what is
100.238?
Confirm the
answer with
your
calculator.
You will be able
to use and prove
the properties of
logarithms
You will be
able to use and prove
the properties of logarithms
Product Property of
Logarithms
Let b, m, and n be
positive real numbers
with b ≠ 1.
logb  m  n  
logb m  logb n
Product Property of
Exponents
bm  bn  bmn
“The log of a product
equals the sum of
the logs of the
factors.”
Prove the Product Property of Logs.
logb  m  n   logb m  logb n
Let logb m  x and logb n  y.
So b x  m and b y  n.
logb  m  n   logb b x  b y

 log b  b 

x y
 x y
 logb m  logb n
Quotient Property of
Logarithms
Let b, m, and n be
positive real numbers
with b ≠ 1.
m
log b   
n
logb m  logb n
Quotient Property of
Exponents
bm
mn
b
n
b
“The log of a quotient
equals the
difference of the
logs of the divisors.”
Power Property of
Logarithms
Let b, m, and n be
positive real numbers
with b ≠ 1.
log b mn 
n  logb m
Power Property of
Exponents
b 
m
n
b
m n
“The log of a number
to a power equals
the power times the
log of the number.”
Use log3 12 ≈ 2.262 and log3 2 ≈ 0.631 to
evaluate the following.
1. log3 6
2. log3 24
3. log3 32
Find the exact value of each expression without
using a calculator.
12
5
1. log 7 5 7
2. ln e  ln e
2
3x
Expand log 7 3
5y
 log 7 3x 2  log 7 5 y 3

 log 7 3  log 7 x  log 7 5  log 7 y
2
3
 log7 3  2log7 x  log7 5  3log7 y

Condense ln8  2ln5  ln10
 ln 8  ln 52  ln10
 ln 8  52  ln10
200
8  52
 ln
 ln 20
 ln
10
10
1. Expand
5 x3
log
y
2. Condense
1
ln 4  ln 3  ln12
3
Use a table of common logs and the properties
of logarithms to multiply 154 x 207.
Use a table of common logs and the properties
of logarithms to divide 375  123.
Use a table of common logs and the properties
of logarithms to evaluate 3613.
You will be able to use and
prove the change of base
formula for logarithms
log 𝜋 3 ?
Recall that a calculator is quite elitist when it comes to
evaluating logs. It prefers the natural or common
logs and deems the rest uncommon or unnatural.
But there’s a way around this electronic prejudice
using this formula:
Let a, b, and c be positive real numbers with b ≠ 1.
log b a
What you’re taking the log of
log c a 
on top and the original base
log b c
on bottom.
Recall that a calculator is quite elitist when it comes to
evaluating logs. It prefers the natural or common
logs and deems the rest uncommon or unnatural.
But there’s a way around this electronic prejudice
using this formula:
Let a, b, and c be positive real numbers with b ≠ 1.
log a
ln a
log c a 
log c a 
log c
ln c
Prove the Change of Base Formula.
So c x  b y
log b a
log c a 
log b c
Let log c a  x
log b a  y
log b c  z
 
b
c a
x
by  a
b c
z
z
x
 by
b zx  b y
zx  y
y
x
z
log b a
log c a 
log b c
Evaluate log6 24 using common and natural logs.
log 24

 1.774
log 6
ln 24

 1.774
ln 6
Use the change of base formula to evaluate the
following.
1. log5 8  1.292
2. log8 14  1.269
3. log26 9  0.674
4. log12 30  1.369
Objectives:
1. To use (and prove)
properties of
logarithms to
simplify expressions
2. To use (and prove)
the change of base
formula
•
•
•
•
•
•
•
Assignment:
P. 243: 9-16 (Some)
P. 243: 17-22 (Some)
P. 243: 23-38 (Some)
P. 243: 39-60 (Some)
P. 244: 61-78 (Some)
P. 245: 93, 94
HW Supplement