Solving Logarithmic Equations and Inequalitie
by CHED on June 16, 2017
lesson duration of 4 minutes
under General Mathematics
generated on June 16, 2017 at 01:08 am
Tags: Inequalities, Logarithmic Equations
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Generated: Jun 16,2017 09:08 AM
Solving Logarithmic Equations and Inequalitie
( 4 mins )
Written By: CHED on May 21, 2016
Subjects: General Mathematics
Tags: Inequalities, Logarithmic Equations
Resources
n/a
Content Standard
The learner demonstrates understanding of key concepts of inverse functions, exponential functions, and logarithmic
functions.
Performance Standard
The learner is able to apply the concepts of inverse functions, exponential functions, and logarithmic functions to
formulate and solve real-life problems with precision and accuracy.
Learning Competencies
The learner solves logarithmic equations and inequalities.
The learner solves problems involving logarithmic functions, equations, and inequalities.
Introduction 1 mins
In solving logarithmic equations and inequalities, it is important for students to remember the restrictions on the values
of b and x in the logarithmic expression
Recall that:
• b can be any positive real number except 1
• x can be any real number
• the entire expression
can be any real number (hence, can be negative).
The previous lessons exposed students not only to a brief encounter with logarithmic equations and inequalities, but
also to simplification techniques involving logarithmic properties.
Remark that in this lesson, they will be able to apply the previous lessons to actually solve for the values that will make
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the statements true. This is to give them a sense of objective and direction.
Lesson Proper 1 mins
(a) Solving logarithmic equations
Some Strategies for solving logarithmic equations
• Rewriting to exponential form
• Using logarithmic properties
• Applying the One-to-One property of logarithmic functions, as stated below.
One-to-One property of Logarithmic Functions
For any logarithmic function
Another useful property is the Zero Factor Property:
EXAMPLE 1. Find the value of x in the following equations.
Solution.
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EXAMPLE 2. Use logarithms to solve for the value of x in the exponential equation 2^x
2^x = 3.
Solution. 2^x
2^x = 3
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(b) Solving logarithmic inequalities
Introduce the topic by asking students to complete the tables below:
Answer:
Ask the following questions:
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We can generalize the observations we made:
Property of Logarithmic Inequalities
Given the logarithmic expression
EXAMPLE 3. Solve the following logarithmic inequalities.
Solution.
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EXAMPLE 4. The 2013 earthquake in Bohol and Cebu had a magnitude of 7.2, while the 2012 earthquake that
occurred in Negros Oriental recorded a 6.7 magnitude. How much more energy was released by the 2013 Bohol/Cebu
earthquake compared to that by the Negros Oriental earthquake? (Refer to Lesson 17 for a discussion of the Richter
scale).
Solution. Let EB and EN be the energy released by the Bohol/Cebu and Negros Oriental earth- quakes, respectively.
We will determine
Based on the given magnitudes,
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EXAMPLE 5. How much more severe is an earthquake with a magnitude of n on a Richter scale, compared to one
with a magnitude of n + 1?
Solution. Let E1 and E2 be the energy released by the earthquakes with magnitude n and n + 1, respectively. We will
determine
These computations indicate that each 1 unit increase in magnitude represents 31.6 times more energy released.
(This result may seem to contradict other sources which state that each 1 unit increase in magnitude represents an
earthquake that is 10 times stronger. However, those computations use amplitude as a measure of strength. The
computations above are based on the energy released by the earthquake).
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EXAMPLE 6. Interest compounded annually
Using the formula A = P (1 + r)^n (Lesson 12, Example 5) where A is the future value of the investment, P is the
principal, r is the fixed annual interest rate, and n is the number of years, how many years will it take an investment to
double if the interest rate per annum is 2.5%?
Solution. Doubling the principal P, we get A = 2P , r = 2.5% = 0.025,
Answer: It will take approximately 28 years for the investment to double.
EXAMPLE 7. (Population growth) The population of the Philippines can be modeled by the function P(x) = 20, 000,
000 · e^0.0251x, where x is the number of years since 1955 (e.g. x = 0 at 1955). Assuming that this model is accurate,
in what year will the population reach 200 million?
Solution.
Answer: Around the year 2046, the Philippine population will reach 200 million.
Trivia: Based on this model, we will reach 100 million in the year 2019. But last July 2014, the Philippines officially
welcomed its 100 millionth baby.6 Hence mathematical models must always be reviewed and verified against new
data.
EXAMPLE 8. In a bacteria culture, an initial population of 5,000 bacteria grows to 12,000 after 90 minutes. Assume
that the growth of bacteria follows an exponential model f(t) = A^ekt representing the number of bacteria after t
minutes. (a) Find A and k, and (b) use the model to determine the number of bacteria after 3 hours.
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Seatwork 1 mins
Seatwork 1. Solve the following logarithmic equations.
Seatwork 2.
Answer:
Seatwork 3. Solve the following logarithmic inequalities.
Individual Activity 1 mins
Ask the students to carefully "dissect" the expression and solve the equation:
Download Teaching Guide Book 0 mins
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