CHAPTER 2
LESSON 5
Teacher’s Guide
The Laws of Logarithms
AW 2.3
MP 2.6
Objective:
• To prove and apply the laws of logarithms
(Note: The calculator screens present in this teacher’s guide do not appear in the student’s version of
lesson 5. It is intended that students will be actively creating their own calculator screens during your
presentation.)
Remember: Logarithms are ____exponents_______.
y
y = log b x means
x=b
y is the exponent necessary to express x as a power of b.
Prelude to the Laws of Logarithms
Simplify the following (without calculators).
10 log 345
(345)
3log3 56
(56)
b logb y
y
The Product Law
Investigation:
On your calculator, compute each of the following:
log12 + log2
log6 + log 4
log8 + log 3
Finally, compute log 24.
So we have the following pattern:
log12 + log 2 = log(12 × 2)



log6 + log4 = log (6 × 4)  = log 24 = 1.380211242


log8 + log3 = log (8 × 3) 
These results illustrate the law of logarithms for multiplication.
logb (m ⋅ n) = log b m + log b n
b > 0, b ≠ 1, m > 0, n > 0
Check:
Compute log2 + log 3 and then log(2 × 3) or log6.
Answer: .7781512504
Proof of the Product Law
Let m and n be any 2 positive numbers. Also assume b > 0 and b ≠ 1.
By the definition of logarithm, we have
m = blog b m and n = blog b n .
(Note: some students will have difficulty understanding this step.)
Then, by substitution, and the product law of powers, we have
m × n = b logb m × blog b n = b( log b m+ log b n)
Finally, by the definition of logarithm, we can say
log b (m × n) = log b m + log b n
Note: This is a good example of what mathematicians call an “elegant” proof: it is
extremely simple, and yet powerful.
Example 1:
Compute without calculator: log 8 32 + log 8 2
Answer = 2
The Quotient Law
Investigation:
On your calculator, compute each of the following:
log 48 − log8
log24 − log 4
log12 − log 2
Finally, compute log 6.
So we have the following pattern.
log48 − log8 = log(48 8) 



log24 − log4 = log(24 4 ) = log6 = .7781512504


log12 − log2 = log(12 2) 
This pattern illustrates the law of logarithms for division.
logb ( m
n ) = log b m - log b n
m > 0, n > 0
Check:
Compute log 45 − log9 and then log( 45 9) or log5.
Answer: .6989700043
Proof of the Quotient Law (Analagous to the proof of the Product Law)
Let m and n be any 2 positive numbers. Also assume b > 0 and b ≠ 1.
By the definition of logarithm, we have
m = blogb m and n = b logb n .
Then, by substitution, and the quotient law of powers, we have
log m
m = b b = b(log b m −log b n)
n
b logb n
Finally, by the definition of logarithm, we can say
log b ( m
n ) = log b m − logb n
The Power Law
Investigation:
On your calculator, compute each of the following:
log 9
2 log 3
log27
log81
3log 3
4log3
Observed pattern:
log9 = log3 2 = 2log3
log27 = log3 3 = 3log3
log81= log34 = 4log3
log243 = log3 5 = 5 log3
This pattern suggests the law of logarithms for powers.
log b m p = p log b m
m > 0, n > 0, b > 0, b ≠ 1, p ∈ ℜ
log243
5log3
Check:
Use your calculator to evaluate log1024 and 10log2.
log1024 = log 210 = 10log2 = 3.0130299957
Proof of the Law of Logarithms for Powers
Let m be any positive real number. Also, assume b > 0 and b ≠ 1.
Then by the definition of logarithms, m = blog b m .
Raising both sides of this equation to the power p, we have
m p = (blog b m ) p .
Using the law of exponents for powers, we can write
m p = b p logb m .
Finally, from the defintion of logarithms (base b), we have the result:
log b m p = p log b m .
This law will prove very useful in the next lesson.
Law of Logarithms for Roots
Since the power law is valid for any real number p, we can state the following corollary
of the power law:
log b r m = 1r log b m
Proof:
1
log b r m = log b (m ) r = 1r log b m
Example 2:
On your calculator, evaluate log 3 500 and 13 log500
Example 3:
Simplify the following without calculators.
log 4 48 + log 4 23 + log4 8
log 4 (16 × 3) + log 4 23 + log 4 (4 × 2)
= [log 4 16 + log 4 3]+ [log 4 2 − log4 3]+ [log 4 4 + log 4 2]
= log4 16 + log 4 2 + log4 4 + log 4 2
= 2 +1/ 2 +1+1/ 2
=4
Example 4:
Given that log 2 a = 5 , evaluate log 2 (4a2 ) .
log 2 (4a 2 )
= log2 4 + log 2a 2
= 2 + 2log 2a
= 2 + 2(5)
= 12
Example 5:
Express as a single logarithm and simplify.
log 8 48 + log 8 4 − log8 3
log 8 48 + log 8 4 − log 8 3


= log8 48 × 4
 3 
= log8 64
=2