At the end of this lesson, the learner should be able to
• accurately apply the laws of logarithms on logarithmic equations;
• correctly solve logarithmic equations.
• correctly solve logarithmic inequalities; and
• correctly apply the laws and properties of logarithms in solving logarithmic inequalities.
Objectives
● How will you apply the laws of logarithms on logarithmic
equations?
● How will you solve logarithmic equations?
Essential Questions
Before we start with our lesson, let’s revisit the laws of logarithms and have a short drill about it by working as a team on the following online activity.
https://www.mathsisfun.com/algebra/exponents-logarithms.html
(Click the link posted in the chat box to access the Jamboardactivity.)
Warm-up!
● What are logarithms?
● What is the relationship between exponents and logarithms?
● What does the Laws of Logarithm state?
Guide Questions
2 Logarithm of a Productthe logarithm of a product is equal to the sum of the logarithms of its factors
Example:
The logarithmic expression log 6 is equal to log 2 + log 3.
4 Logarithm of a Quotient the logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator
Example:
The logarithmic expression log2
3is equal to log 2 − log 3.
Example:
The logarithmic expression log 8 is equal to 3 log 2.
6 Logarithm of a Powerthe logarithm of a power 𝑚𝑛 to the base 𝑏 is equal to the product of the exponent 𝑛 and the logarithm of 𝑚 to the base 𝑏
Logarithmic Equationan equation that involves logarithmic expressions; the variable is written as part of the argument of the logarithm
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Examples:
a. log3 𝑥 + 1 = 2b. log3 𝑥2 − 8 − log3 4 = 3c. ln 2𝑥 + 3 = ln 4𝑥
Rule Property
One-to-one correspondence If log𝑎 𝑥 = log𝑎 𝑦, then 𝑥 = 𝑦.
Identity property If log𝑎 𝑎 = 𝑦, then 𝑦 = 1.
Zero logarithm If log𝑎 1 = 𝑦, then 𝑦 = 0.
Some Properties of Logarithms8
Rule Property
Natural logarithm If ln 𝑥 = 𝑦, then log𝑒 𝑥 = 𝑦.
Equivalent exponential form If log𝑎 𝑥 = 𝑦, then 𝑎𝑦 = 𝑥.
Change of base If log𝑎 𝑥 = 𝑦 , then log𝑏 𝑥
log𝑏 𝑎= 𝑦.
Some Properties of Logarithms8
Example 1: Solve the logarithmic equation log2 𝑥 = 3.
Solution:We use the definition of logarithms. Recall that if log𝑎 𝑥 = 𝑦,then 𝑎𝑦 = 𝑥.
log2 𝑥 = 3
𝑥 = 23
𝑥 = 8
Hence, the solution is 𝒙 = 𝟖.
Example 2: Find the solution(s) of the logarithmic equation
ln 3𝑥 − 2 + ln 4 = ln(𝑥 + 3).
Solution:1. Apply the Addition Law of Logarithms on the left-hand
side of the equation.
ln 3𝑥 − 2 4 = ln(𝑥 + 3)
Example 2: Find the solution(s) of the logarithmic equation
ln 3𝑥 − 2 + ln 4 = ln(𝑥 + 3).
Solution:2. Apply the one-to-one correspondence property of
logarithms.
3𝑥 − 2 4 = 𝑥 + 3
Example 2: Find the solution(s) of the logarithmic equation
ln 3𝑥 − 2 + ln 4 = ln(𝑥 + 3).
Solution:3. Solve the resulting linear equation.
3𝑥 − 2 4 = 𝑥 + 312𝑥 − 8 = 𝑥 + 3
11𝑥 = 11𝑥 = 1
Example 2: Find the solution(s) of the logarithmic equation
ln 3𝑥 − 2 + ln 4 = ln(𝑥 + 3).
Solution:Therefore, the solution is 𝒙 = 𝟏.
Logarithmic Inequalityan inequality that contains logarithmic expressions
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Examples:
a. log4 𝑥 + 8 ≥ 11b. log(𝑥 − 2) < log 𝑥2 − 4c. 5 + ln 2𝑥 > 𝑒
One-to-One Correspondence Rule The following rules help in solving logarithmic inequalities.
a. log𝑏 𝑥 > log𝑏 𝑦 if and only if 𝑥 > 𝑦 and 𝑏 > 1.
b. log𝑏 𝑥 < log𝑏 𝑦 if and only if 𝑥 > 𝑦 and 0 < 𝑏 < 1.
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Examples:
log2 32 > log2 16 since 32 > 16 and 𝑏 = 2 > 1.
Example 1: Solve the inequality log2(2𝑥 + 3) > log2 3𝑥.
Solution:We can apply the one-to-one correspondence rule in the inequality. Since log2(2𝑥 + 3) > log2 3𝑥, and the bases of the logarithms on each side of the inequality are equal and greater than 1, it follows that 𝟐𝒙 + 𝟑 > 𝟑𝒙.
Example 1: Solve the inequality log2(2𝑥 + 3) > log2 3𝑥.
Solution:Solve the resulting linear inequality.
2𝑥 + 3 > 3𝑥3 > 3𝑥 − 2𝑥3 > 𝑥
Example 1: Solve the inequality log2(2𝑥 + 3) > log2 3𝑥.
Solution:Therefore, 𝑥 < 3. However, since 2𝑥 + 3 and 3𝑥 are arguments in the original inequality, it means that 2𝑥 + 3 > 0and 3𝑥 > 0.
Solving for 𝑥, we get 𝑥 > −3
2and 𝑥 > 0.
Therefore, the solution set is 𝟎 < 𝒙 < 𝟑.
Example 2: Solve the inequality log3 𝑥 − 2 < 5.
Solution:1. Rewrite the right-hand side of the inequality as 𝐥𝐨𝐠𝟑 𝟑
𝟓 so that it becomes a logarithm with the same base as the logarithm on the left-hand side.
Note that log3 35 = 5 log3 3 = 5 1 = 5.
log3 𝑥 − 2 < log3 35
Example 2: Solve the inequality log3 𝑥 − 2 < 5.
Solution:2. Apply the one-to-one correspondence rule on the
inequality.
Since log3(𝑥 − 2) < log3 35 and the bases of the logarithms on
each side of the inequality are equal and greater than 1, it follows that 𝒙 − 𝟐 < 𝟑𝟓.
Example 2: Solve the inequality log3 𝑥 − 2 < 5.
Solution:Solve the resulting linear inequality.
𝑥 − 2 < 35
𝑥 − 2 < 243𝑥 < 245
Example 2: Solve the inequality log3 𝑥 − 2 < 5.
Solution:Therefore, 𝑥 < 245. However, since 𝑥 − 2 is an argument in the original inequality, it means that 𝑥 − 2 > 0.
Solving for 𝑥, we get 𝑥 > 2.
Therefore, the solution set is 𝟐 < 𝒙 < 𝟐𝟒𝟓.
Individual Practice:
1. Solve the logarithmic equation log3(2𝑥 + 1) = 2.
2. Solve the logarithmic equation log3 𝑥 + log5 𝑥 = 3.
Group Practice: To be done in groups of four.
If a certain bacteria double in number every day, how long will it take for 1 000 bacteria to reach 16 384 000 in number?
Group Practice: To be done in groups of three or four.
Interns Maggie and Michelle are appointed to two different positions in a company. Maggie’s allowance (in thousand pesos) is log6(𝑥 − 2) while Michelle’s allowance (in thousand pesos) is log5(𝑥 − 2). Find the possible values of 𝑥 if Michelle’s allowance is higher than that of Maggie’s.