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Properties of Logarithms
Section 3.3
Objectives
• Rewrite logarithms with different bases.
• Use properties of logarithms to evaluate or
rewrite logarithmic expressions.
• Use properties of logarithms to expand or
condense logarithmic expressions.
History of Logarithms
John Napier, a 16th Century
Scottish scholar, contributed
a host of mathematical
discoveries.
John Napier (1550 – 1617)
He is credited with creating the
first computing machine,
logarithms and was the first to
describe the systematic use of the
decimal point.
Other contributions include a
mnemonic for formulas used in
solving spherical triangles and two
formulas known as Napier's
analogies.
“In computing tables, these large numbers may again
be made still larger by placing a period after the
number and adding ciphers. ... In numbers
distinguished thus by a period in their midst,
whatever is written after the period is a fraction, the
denominator of which is unity with as many ciphers
after it as there are figures after the period.”
Napier lived during a time
when revolutionary
astronomical discoveries
were being made.
Copernicus’ theory of the
solar system was published in
1543, and soon astronomers
were calculating planetary
positions using his ideas.
But 16th century arithmetic
was barely up to the task and
Napier became interested in
this problem.
Nicolaus Copernicus (1473-1543)
Even the most basic
astronomical arithmetic
calculations are ponderous.
Johannes Kepler (1571-1630)
filled nearly 1000 large pages
with dense arithmetic while
discovering his laws of planetary
motion!
A typical page from one of
Kepler’s notebooks
Johannes Kepler (1571-1630)
Napier’s Bones
In 1617, the last year of his life,
Napier invented a tool called
“Napier's Bones” which reduces the
effort it takes to multiply numbers.
“Seeing there is nothing that is so troublesome to
mathematical practice, nor that doth more molest and
hinder calculators, than the multiplications, divisions...
I began therefore to consider in my mind by what
certain and ready art I might remove those hindrances.”
Logarithms Appear
The first definition of the logarithm was constructed by
Napier and popularized by a pamphlet published in 1614,
two years before his death. His goal: reduce multiplication,
division, and root extraction to simple addition and
subtraction.
Napier defined the "logarithm" L of a number N by:
N==107(1-10(-7))L
This is written as NapLog(N) = L or NL(N) = L
While Napier's definition for logarithms is different from the
modern one, it transforms multiplication and division into
addition and subtraction in exactly the same way.
Logarithmic FAQs
• Logarithms are a mathematical tool originally invented to
reduce arithmetic computations.
• Multiplication and division are reduced to simple addition
and subtraction.
• Exponentiation and root operations are reduced to more
simple exponent multiplication or division.
• Changing the base of numbers is simplified.
• Scientific and graphing calculators provide logarithm
functions for base 10 (common) and base e (natural) logs.
Both log types can be used for ordinary calculations.
Logarithmic Notation
• For logarithmic functions we use the
notation:
loga(x) or logax
• This is read “log, base a, of x.” Thus,
y = logax means x = ay
• And so a logarithm is simply an exponent
of some base.
Change-of-Base Formula
Only logarithms with base 10 or base e can be found by using a
calculator. Other bases require the use of the Change-of-Base
Formula.
Change-of-Base Formula
If a  1, and b  1, and M are positive real numbers, then
logb M
log a M 
.
logb a
Example:
Approximate log4 25.
log4 25 
10 is used for
both bases.
log10 25 log 25 1.39794  2.32193


log10 4 log 4 0.60206
The Change-of-Base Rule
Change-of-Base Rule
For any positive real numbers x, a, and b,
where a  1 and b  1, log x  log b x .
a
log b a
Proof
Let
y  log a x.
ay  x
log b a y  log b x
y log b a  log b x
log
y  log
x
ba
b
log
 log a x  log
x
ba
b
Change of base formula:
• u, b, and c are positive numbers with b≠1 and
c≠1. Then:
log b u
• logcu =
log b c
log u
• logcu =
log c
(base 10)
ln u
• logcu =
ln c
(base e)
Change-of-Base Formula
Example:
Approximate the following logarithms.
(a) log3 198
log3 198 
log198 2.297

 4.816
log 3
0.477
(b) log6 5
log6
log 5  0.349
 0.449
5
0.778
log 6
Examples:
• Use the change of base to evaluate:
• log37 =
• (base 10)
•(base e)
• log 7 ≈
• log 3
• 1.771
•ln 7 ≈
•ln 3
•1.771
Your Turn:
Evaluate each expression and round to four
decimal places.
(a) log 5 17
Solution
(a) 1.7604
(b) -3.3219
(b) log 2 .1
Properties of Logarithms
For x > 0, y > 0, a > 0, a  1, and any real
number r,
Product Rule log a xy  log a x  log a y.
Quotient Rule log a xy  log a x  log a y.
r
log
x
 r log a x.
Power Rule
a
Examples Assume all variables are positive.
Rewrite each expression using the properties
of logarithms.
1. log 8 x  log 8  log x
15
2. log9  log9 15  log9 7
7
1
1
2
3. log5 8  log5 8  log5 8
2
The Product Rule of Logarithms
Product Rule of Logarithms
If M, N, and a are positive real numbers, with
a  1, then loga(MN) = logaM + logaN.
Example: Write the following logarithm as a sum of
logarithms.
(a) log5(4 · 7)
log5(4 · 7) = log54 + log57
(b) log10(100 · 1000)
log10(100 · 1000) = log10100 + log101000
=2+3=5
Your Turn:
• Express as a sum of
logarithms:
2
log3 ( x w)
Solution:
log3 ( x w)  log3 x  log3 w
2
2
The Quotient Rule of Logarithms
Quotient Rule of Logarithms
If M, N, and a are positive real numbers, with
a  1, then log  M   log M  log N.
a

N 
a
a
Example: Write the following logarithm as a
difference of logarithms.
10
(a) log5   = log5 10  log5 3
 3
 c
(b) log8    log8 c  log8 4
 4
Your Turn:
• Express as a difference
of logarithms.
10
log a
b
• Solution:
10
log a  log a 10  log a b
b
Sum and Difference of Logarithms
 8y
log
Example: Write
as the sum or difference
6
 5 
of logarithms.
 8y
log6    log 6(8 y)  log 6 5
Quotient Rule
 5
 log 6 8  log 6 y  log 6 5 Product Rule
The Power Rule of Logarithms
The Power Rule of Logarithms
If M and a are positive real numbers, with a
 1, and r is any real number,
then loga M r = r loga M.
Example: Use the Power Rule to express all
powers as factors.
log4(a3b5) = log4(a3) + log4(b5)
= 3 log4a + 5 log4b
Product Rule
Power Rule
Your Turn:
• Express as a product.
log a 7
3
Solution:
3
log a 7  3log a 7
Your Turn:
• Express as a product.
5
log a 11
• Solution:
log a 11  log a 11
5
1/5
1
 log a 11
5
NOT
Laws of Logarithms
Warning
Rewriting Logarithmic Expressions
• The properties of logarithms are useful for rewriting
logarithmic expressions in forms that simplify the
operations of algebra.
• This is because the properties convert more
complicated products, quotients, and exponential
forms into simpler sums, differences, and products.
• This is called expanding a logarithmic expression.
• The procedure above can be reversed to produce a
single logarithmic expression.
• This is called condensing a logarithmic
expression.
Examples:
• Expand:
• log 5mn =
• log 5 + log m + log n
• Expand:
• log58x3 =
• log58 + 3·log5x
Expand – Express as a Summ and
Difference of Logarithms
7x
• log2 =
y
3
• log27x3 - log2y =
• log27 + log2x3 – log2y =
• log27 + 3·log2x – log2y
Condense - Express as a Single
Logarithm
Example: Write the following as the
logarithm of a single expression.
5log6(x  3)  [2log6(x  4)  3log 6 x]
5log6(x  3)  [2log6(x  4)  3log 6 x]
 log 6(x  3)5  [log 6(x  4) 2  log 6 x3]
Power Rule
 log 6(x  3)5  [log 6(x  4) 2  x3]
Product Rule
 (x  3)5 
 log 6 
2 3
(
x

4)
x 

Quotient Rule
Condensing Logarithms
• log 6 + 2 log2 – log 3 =
• log 6 + log 22 – log 3 =
• log (6·22) – log 3 =
• log 6  2 =
2
3
• log 8
Examples:
• Condense:
• log57 + 3·log5t =
• log57t3
• Condense:
• 3log2x – (log24 + log2y)=
3
x
• log2
4y
Your Turn:
• Express in terms of sums and
differences of logarithms.
3
wy
log a 2
z
4
• Solution:
3
4
w y
3 4
2
log a 2  log a ( w y )  log a z
z
 log a w3  log a y 4  log a z 2
 3log a w  4log a y  2log a z
Your Turn:
Assume all variables are positive. Use the
properties of logarithms to rewrite the expression
log
3 5
n x y
b
zm
.
Solution logb n
x y
x y 

log
b
m
m 
z
 z 
3 5
3 5
1
n
3 5
x
y
 1 logb m
n
z
 1 logb x 3  logb y 5 logb z m
n
 1 3 logb x 5 logb y  m logb z 
n
 3 logb x  5 logb y  m logb z
n
n
n


Your Turn:
• Express as a single logarithm.
1
6log b x  2log b y  log b z
3
• Solution:
1
6logb x  2logb y  logb z  logb x 6  logb y 2  logb z1/3
3
x6
1/3
 logb 2  log b z
y
x 6 z1/3
x6 3 z
 logb 2 , or log b 2
y
y
Your Turn:
Use the properties of logarithms to write
2
3
1
log
m

log
2
n

log
m
n as a single
b
b
b
2
2
logarithm with coefficient 1.
Solution 12 log b m  32 log b 2n  log b m 2 n
 log b m  log b 2n  2  log b m 2 n
1
3
2
2


m
2
n
 log b
m2n
3
1
2
2
2
n
 log b
3
m 2
1
3
2


2
n
 log b  3   log b 8n3
m 
m
1
2
3
Another Type of Problem
• If loga3 = x and loga4 = y,
express each log expression in
terms of x and y.
1. loga12
•
Loga(3•4) = loga3 + loga4 = x+y
2. Log34
•
Log34 = loga4/loga3 = y/x
Assignment
• Pg. 211 – 213: #1 – 19 odd, 31, 33, 37 –
55 odd, 59 – 75 odd