Download Lesson 6 Laws of Logarithms

LESSON 6: LAWS OF LOGARITHMS I
Learning Outcomes:

To develop the laws of logarithms

To determine an equivalent form of a logarithmic expression using the laws of
logarithms
Investigating the Laws of Logarithms
1. Show that log (1000 x 100) ≠ (log 1000)(log 100).
2. Evaluate the following:
a. i. log 6 + log 5
ii. Log 30
b. i. log 7 + log 3
ii. Log 21
3. Based on the results in the previous section, suggest a possible law for log M + log N, where
M and N are positive numbers.
4. Use your conjecture to express log 1000 + log 100 as a single logarithm.
5. Show that 𝑙𝑜𝑔
1000
100
≠
𝑙𝑜𝑔1000
𝑙𝑜𝑔100
.
6. Evaluate the following:
a. i. log 12
ii. log 48 – log 4
b. i. log 7
ii. log 35 – log 5
7. Based on the results in the previous section, suggest a possible law for log M - log N, where M
and N are positive numbers.
8. Use your conjecture to express log 1000 – log 100 as a single logarithm.
9. Show that log 10002 ≠ (log 1000)2
10. Evaluate the following:
a. i. 3 log 5
ii. log 125
b. i. 4 log 2
ii. log 16
11. Based on the results in the previous section, suggest a possible law for Plog M, where M and
N are positive numbers.
12. Use your conjecture to express 2 log 1000 as a logarithm without a coefficient
13. The laws of common logarithms are also true for any logarithm with a base that is a positive
real number other than 1. Without technology, evaluate each of the following:
a. 𝑙𝑜𝑔6 18 + 𝑙𝑜𝑔6 2
b. 𝑙𝑜𝑔2 40 − 𝑙𝑜𝑔2 5
c. 4𝑙𝑜𝑔9 3
Since logarithms are exponents, the laws of logarithms are related to the laws of powers
Product Law of Logarithms
𝑙𝑜𝑔𝑐 𝑀𝑁 = 𝑙𝑜𝑔𝑐 𝑀 + 𝑙𝑜𝑔𝑐 𝑁
Proof:
Let 𝑙𝑜𝑔𝑐 𝑀 = 𝑥 and 𝑙𝑜𝑔𝑐 𝑁 = 𝑦.
Turn each equation into exponential form as 𝑀 = 𝑐 𝑥 and 𝑁 = 𝑐 𝑦 .
𝑀𝑁 = (𝑐 𝑥 )(𝑐 𝑦 )
𝑀𝑁 = 𝑐 𝑥+𝑦
Write in logarithmic form
𝑙𝑜𝑔𝑐 𝑀𝑁 = 𝑥 + 𝑦
𝑙𝑜𝑔𝑐 𝑀𝑁 = 𝑙𝑜𝑔𝑐 𝑀 + 𝑙𝑜𝑔𝑐 𝑁
Substitute for x and y
Quotient Law of Logarithms
𝑙𝑜𝑔𝑐
𝑀
= 𝑙𝑜𝑔𝑐 𝑀 − 𝑙𝑜𝑔𝑐 𝑁
𝑁
Proof
Let 𝑙𝑜𝑔𝑐 𝑀 = 𝑥 and 𝑙𝑜𝑔𝑐 𝑁 = 𝑦.
Turn each equation into exponential form as 𝑀 = 𝑐 𝑥 and 𝑁 = 𝑐 𝑦
𝑀 𝑐𝑥
=
𝑁 𝑐𝑦
𝑀
𝑁
= 𝑐 𝑥−𝑦
𝑀
𝑙𝑜𝑔𝑐 𝑁 = 𝑥 − 𝑦
Write in logarithmic form
𝑀
𝑙𝑜𝑔𝑐 𝑁 = 𝑙𝑜𝑔𝑐 𝑀 − 𝑙𝑜𝑔𝑐 𝑁
Substitute for x and y
Power Law of Logarithms
𝑙𝑜𝑔𝑐 𝑀𝑃 = 𝑃𝑙𝑜𝑔𝑐 𝑀
Proof
Let 𝑙𝑜𝑔𝑐 𝑀 = 𝑥, where M and c are positive real numbers with c≠1.
Write the equation in exponential form as 𝑀 = 𝑐 𝑥
Let P be a real number.
𝑀 = 𝑐𝑥
𝑀𝑃 = (𝑐 𝑥 )𝑃
Substitute in power law
(multiply by an exponent)
𝑀𝑃 = 𝑐 𝑥𝑃
𝑙𝑜𝑔𝑐 𝑀𝑃 = 𝑥𝑃
𝑙𝑜𝑔𝑐 𝑀𝑃 = (𝑙𝑜𝑔𝑐 𝑀)𝑃
𝑙𝑜𝑔𝑐 𝑀𝑃 = 𝑃𝑙𝑜𝑔𝑐 𝑀
Write in logarithmic form
Substitute for x and y
Ex. Write each expression in terms of individual logarithms of x, y and z.
𝑥
a. 𝑙𝑜𝑔6 𝑦
b. 𝑙𝑜𝑔5 √𝑥𝑦
c. 𝑙𝑜𝑔3 3
9
√𝑥 2
d. 𝑙𝑜𝑔7
𝑥5𝑦
√𝑧
Assignment: pg. 400-403 #1-6