Chapter 8 – Properties of Logarithms and Natural Logarithm PDF

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Summary

Logarithms were actually discovered and used in ancient times by both Indian and
Islamic mathematicians. They were not used widely, though, until the 1600’s, when
logarithms simplified the large amounts of hand computations needed in the scientific
explorations of the times. In par...

Description

Chapter 8 – Properties of Logarithms and Natural Logarithm In this lesson, we continue the study of logarithms. We explore some properties of logarithms. They have many uses; we will use them to solve some equations. We also examine another special logarithm, called the natural logarithm and practice with it through applications.

Lesson Topics: Section 8.1 Properties of Logarithms ! ! !

Properties of Logarithms Practice with Properties of Logarithms Solving Equations

Section 8.2 Natural Logarithm ! ! !

Compound Interest The Natural Base and Applications The Natural Logarithm and Applications

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Lesson 8 Section 8.1 – Properties of Logarithms In Lesson 7, we learned about logarithms. Similar to the Properties of Exponents we studied in Lesson 4, logarithms have certain key properties. They are: If 𝑀 and N are positive real numbers: Product Rule

𝐥𝐨𝐠 𝒃 𝑴 ∙ 𝑵 = 𝐥𝐨𝐠𝒃 𝑴 + 𝐥𝐨𝐠 𝒃 𝑵

Quotient Rule

𝐥𝐨𝐠 𝒃

Power Rule

𝐥𝐨𝐠 𝒃 𝑵𝒑 = 𝒑 ∙ 𝐥𝐨𝐠𝒃 𝑵

𝑴 𝑵

= 𝐥𝐨𝐠 𝒃 𝑴 − 𝐥𝐨𝐠𝒃 𝑵

While we will not prove these properties, the following examples help get an idea about why they are true. For the purposes of demonstration, let 𝑀 = 32, 𝑁 = 4, 𝑏 = 2!𝑎𝑛𝑑 !𝑝 = 3. Let’s look at each of the rules. 1)

Product Rule:

Is

Evaluate the left hand side:

log ! 32 ∙ 4 = log ! 32 + log ! 4? log ! 32 ∙ 4 = ___________________________________

Evaluate the right hand side: log ! 32 + log ! 4 = _______________________________ In this example, logarithms are exponents on base 2. For 32 and 4, the corresponding exponents are 5 and 2. Hence, when we multiply 32 and 4, their exponents, base 2, add! That is, 32 ∙ 4 = 2! ∙ 2! = 2!!! = 2! . This is logarithmic version of the Product Rule for Exponents.

2)

Quotient Rule:

Is

Evaluate the left hand side:

log ! log !

!" ! !" !

= log ! 32 − log ! 4? = _____________________________________

Evaluate the right hand side: log ! 32 − log ! 4 = _______________________________ The division of numbers with the same base corresponds to subtracting the exponents. Hence, !" !

!!

= !! = 2!!! = 2! . This is logarithmic version of the Quotient Rule for Exponents.

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Lesson 8 – Properties of Log and Natural Log

3)

Power Rule:

Is

Evaluate the left hand side:

Lesson

log ! 4! = 3 ∙ log ! 4? log ! 4! = ______________________________________

Evaluate the right hand side: 3 ∙ log ! 4 = _____________________________________ Here, from an exponential point of view, 4! = 2! version of the Power Rule for Exponents. Problem 1

!

= 2! . So, this rule is the logarithmic

WORKED EXAMPLE – Expanding Logarithms

Fully expand each of the following logarithms using the properties of logarithms. In your final answer, the argument in each logarithmic term should be a single variable. a) log ! 9𝑥 !

b) log !

!" !

= log ! 9 + log ! 𝑥 !

= log ! 81 − log ! 𝑎

= 1 + 2 log ! 𝑥

= 4 − log ! 𝑎

Problem 2 a) log !

Problem 3

IN-CLASS EXAMPLE – Expanding Logarithms !"#

b) log

!"

! ! !

!

YOU TRY – Expanding Logarithms

Fully expand each of the following logarithms using the properties of logarithms. In your final answer, the argument in each logarithmic term should be a single variable. !"! a) log ! 32𝑎! 𝑏 ! b) log ! !

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Lesson 8 – Properties of Log and Natural Log Problem 4

Lesson

WORKED/IN-CLASS EXAMPLE – Condensing Logarithms

Use the properties of logarithms to condense each logarithmic expression. Your final answer should be an expression that is a single logarithm with coefficient 1. a) log 𝑥 + 7log 𝑦

b) 3 log ! 𝑎 + 2 log ! 𝑏 − log ! 𝑐

= log 𝑥 + log 𝑦 ! = log 𝑥𝑦 !

Problem 5

YOU TRY – Condensing Logarithms

Use the properties of logarithms to condense each logarithmic expression. Your final answer should be an expression that is a single logarithm with coefficient 1. a) 2!log 𝑎 − log 𝑏

b) log ! 𝑥 − 2 log ! 𝑦 + 4!log ! 𝑧

Solving Equations revisited… Recall that we learned to solve logarithmic equations in Lesson 7. In some cases, it’s useful to apply the properties of logarithms to simplify one or both sides of an equation before solving it. Since you can only take the logarithm of a positive number, you must check for extraneous solutions. That is, be sure to check that each “solution” you come up is actually a solution by substituting it into the original equation to make sure it satisfies it.

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Lesson 8 – Properties of Log and Natural Log Problem 6

Lesson

WORKED EXAMPLE – Solving Logarithmic Equations

Solve the following equation using properties of logarithms as needed. Be sure to check for extraneous solutions. log ! 𝑥 − 5 + log ! 𝑥 + 3 ! = !2 log ! 𝑥 − 5 𝑥 + 3 ! = !2 3! = ! 𝑥 − 5 𝑥 + 3 𝑥 ! − 2𝑥 − 15 = 9 𝑥 ! − 2𝑥 − 24 = 0 𝑥−6 𝑥+4 =0 𝑥 − 6 = 0 or 𝑥 + 4 = 0 𝑥 = 6 or 𝑥 = −4

Using the Product Rule Converting to exponential form Multiplying out Preparing to solve the quadratic equation Factoring One of the factors must be 0

Checking solutions:

So,

𝑥 = 6: log ! 6 − 5 + log ! 6 + 3 = log ! 1 + log ! 9 =0+2 =2

𝑥 = −4: log ! −4 − 5 + log ! −4 + 3 = log ! −9 + log ! −1 neither expression is defined since we cannot take the log of a negative number

𝒙 = 𝟔 is a solution, while

𝒙 = −𝟒 is not a solution.

Problem 7 IN-CLASS EXAMPLE – Solving Logarithmic Equations Solve the following equation using properties of logarithms as needed. Be sure to check for extraneous solutions. log ! 10𝑥 − 8 − log ! 𝑥 ! = !2

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Lesson 8 – Properties of Log and Natural Log Problem 8

Lesson

YOU TRY – Solving Logarithmic Equations

Solve each of the following equations, using properties of logarithms as needed. Be sure to check for extraneous solutions. a)

log ! 𝑥 + 5 + log ! 𝑥 ! = !2

b)

log ! 𝑥 + 2 − log ! 𝑥 − 5 ! = !3

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Lesson 8 – Properties of Log and Natural Log

Lesson

Section 8.2 – The Natural Logarithm To motivate what is called the “natural base” in mathematics, let’s consider how money grows in an account when interest is compounded. Suppose you invest $1000 in an account at 3% annual interest for 𝑡 years. As we learned in Lesson 6, the amount of money in such an account after 𝑡 years would be given by 𝐴 𝑡 = 1000 1 + 0.03

!

Now, one could compound the interest more than one time per year. For example, interest could be compounded: • • • • •

Semi-annually, or 2 times per year Quarterly, or 4 times per year Monthly, or 12 times per year Weekly, or 52 times per year Daily, or 365 times per year

For semi-annual compounding in our example, the 3% annual interest would be divided into two equal parts, or 1.5%, and that rate would be applied twice a year. So, if $1000 is invested at 3% annual interest, compounded semi-annually, after 𝑡 years, the amount of money in such an account would be given by 0.03 !! 𝐴 𝑡 = 1000 1 + 2 Note that the exponent is 2𝑡 since the interest is applied twice a year. So, over 𝑡 years, the interest would be applied 2𝑡 times. Applying similar logic to quarterly, monthly and daily compounding in our example, the amount in the account after 𝑡 years would be given by Semi-annual compounding:

𝐴 𝑡 = 1000 1 +

Quarterly compounding:

𝐴 𝑡 = 1000 1 +

Monthly compounding:

𝐴 𝑡 = 1000 1 +

Weekly compounding:

𝐴 𝑡 = 1000 1 +

Daily compounding:

𝐴 𝑡 = 1000 1 +

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!.!" !! ! !.!" !! ! !.!" !"! !" !.!" !"! !" !.!" !"#! !"#

Lesson 8 – Properties of Log and Natural Log

Lesson

You may wonder: Does it make a difference how many times we compound per year, since at the end of the year, you’ve still applied 3%? Let’s see. Suppose we invest $1000 at 3% annual interest for 5 years, but with different number of compounding periods.

Compounded (number of times)

Amount after 5 years !

!!!!!!!! = $1159.27

Annually (1 time per year)

1000 1 + 0.03

Semi-Annually (2 times per year)

1000 1 +

Quarterly (4 times per year)

1000 1 +

0.03 1000 1 + 12

!"∙!

Monthly (12 times per year)

1000 1 +

0.03 52

!"∙!

Weekly (52 times per year)

1000 1 +

0.03 365

!"#∙!

Daily (365 times per year)

!.!" !∙! !

!!!!!! = $1160.54

!.!" !∙! !!!!!! = !

$1161.18

!! = $1161.62 !! = $1161.78 = $1161.83

What appears to happen to the amount of money after 5 years as we increase the number of compounding periods? It approaches a certain upper limit, given by 1000 ∙ 𝑒 !.!"

!

= 1161.83

You can compute this quantity by finding the “e” or the “𝑒 ! ” on your calculator. In general, it turns out that, as 𝑛, the number of compounding periods increase, 𝐴 𝑡 =𝑃 1+

! !" !

as!!!becomes'larger

𝐴 𝑡 = 𝑃𝑒 !"

where 𝑒 is a very special irrational number, called Euler’s constant, with 𝑒 ≈ 2.7182818284 … In the world of compound interest, this is called continuous compounding. As in the above example, 𝑒 shows up unexpectedly in many areas of mathematics. Similar to 𝜋, it’s a special number in mathematics, having many unusual properties. Therefore, it is sometimes referred to as the natural base. It’s a base that is used in describing continually growing processes. In the above example, we saw it in the context of compound interest, but it can also be used to model other processes, such as population growth and radioactive decay.

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Lesson 8 – Properties of Log and Natural Log Problem 9

Lesson

IN-CLASS EXAMPLE – Modeling with natural base

The population of California, in millions of people, can be modeled by 𝑃 𝑡 = 37.35𝑒 !.!!"#! , where 𝑡 is the number of years since 2010. a)

What does the coefficient 37.35 represent in the context of the problem?

b)

Use the model to estimate the population of California in the year 2020.

Problem 10

YOU TRY – Modeling with natural base

The radioactive substance Iodine-131 is used to treat thyroid cancer. The half-life of Iodine-131 is about 8 days. Suppose a thyroid cancer patient is given a dosage containing 100 millicuries of Iodine-131. Then, the amount of Iodine-131 remaining in the patient’s bloodstream 𝑡 days after receiving the drug is given by 𝐼 𝑡 = 100𝑒 !!.!"##! . a) How much Iodine-131 would remain in the patient’s body 5 days after they are treated with the drug?

b) What does the statement “The half-life of Iodine-131 is about 8 days” mean? Check with calculation that the equation for the model is consistent with this statement.

Since 𝑒 is often used as a base in exponential models, when solving exponential equations with that base, we would need to use logarithm with base 𝑒. For brevity, the logarithm with base 𝑒 is simply called the natural logarithm. The notation is log ! 𝑥 = ln 𝑥

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Lesson 8 – Properties of Log and Natural Log Problem 11

Lesson

YOU TRY – Practice with the natural logarithm

Evaluate each of the following if possible. If not possible, explain why not. a)

ln 𝑒

b)

ln 𝑒 !

d)

ln −𝑒

e)

ln

Problem 12

! !

c)

ln 1

f)

ln 0

WORKED EXAMPLE – Solving exponential equations with natural base

Solve each of the following equations. Give your answer in EXACT form and as an approximation rounded to three decimal places. Check your answer using the calculator. a)

𝑒 !!!!! = 3

b)

5𝑥 + 1 = 𝑒 ! 5𝑥 = 𝑒 ! − 1

−2𝑥 + 1 = ln 3 −2𝑥 = ln 3 − 1 𝑥=

!" !!!

𝑥=

!!

Check: !!.!"# !!

! ! !! !

𝑥 ≈ 3.817

𝑥 ≈ −0.049

𝑒 !!

ln 5𝑥 + 1 = 3

Check: ln 5 3.817 + 1 ≈ 3

≈3

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Lesson 8 – Properties of Log and Natural Log

Problem 13

Lesson

IN-CLASS EXAMPLE – Solving exponential equations with natural base

Solve each of the following equations. Give your answer in EXACT form and as an approximation rounded to three decimal places. Check your answer using the calculator. a)

1.5𝑒 ! − 4.3 = 12.1

Problem 14

b)

3 ln 2 + 0.3𝑥 − 4 = 5.9

YOU TRY – Solving exponential equations with natural base

Solve each of the following equations. Give your answer in EXACT form and as an approximation rounded to three decimal place. a)

𝑒 !.!"! + 5.7 = 32.2

b)

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ln 0.2𝑥 − 14 = −9.3

Lesson 8 – Properties of Log and Natural Log Problem 15

Lesson

IN-CLASS EXAMPLE – Solving equations with natural base Application

The population of California, in millions of people, can be modeled by 𝑃 𝑡 = 37.35𝑒 !.!!"#! , where 𝑡 is the number of years since 2010. a) Estimate the year in which the population of California will be 60 million, according to this model.

b) Make a good graph of 𝑃 𝑡 = 37.35𝑒 !.!!"#! and show your solution to part a) in your graph. Also, check your solution using the calculator.

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Lesson 8 – Properties of Log and Natural Log Problem 16

Lesson

YOU TRY – Solving exponential equations with natural base Application

Suppose a thyroid cancer patient is given a dosage containing 100 millicuries of Iodine-131. The amount of Iodine-131 remaining in the patient’s bloodstream 𝑡 days after receiving the drug is given by 𝐼 𝑡 = 100𝑒 !!.!"##! . a) After how many days will 10 millicuries of Iodine-131 remain in this patient’s body?

b) Make a good graph of 𝐼 𝑡 = 100𝑒 !!.!"##! and show your solution to part a) on your graph. Also, check your solution using the calculator.

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Name _________________________________

Date ____________________

Chapter 8 Practice Problems Section 8.1: Properties of Logarithms 1. Evaluate each of the following logarithms without the use of a calculator. ! b) log !" 1 a) log ! !"

2.

c)

log log 10

d)

log 10000

e)

log ! 0.25

f)

log 0.0001

Evaluate without using the calculator. You will find the properties of logarithms useful. a) log ! 32 ∙ 16 b) log ! 7!

c)

3.

log !

!! !

Fully expand each of the following logarithms using the properties of logarithms. In your final answer, the argument in each logarithmic term should be a single variable. !" b) log ! 125𝑥 ! 𝑦 ! a) log ! !"

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Chapter 8 – Properties of Log and Natural Log

c)

4.

log

!"! !

Assessment

d)

! !"

log !

!! !

!

!

Use the properties of logarithms to condense each logarithmic expression. Your final answer should be an expression that is a single logarithm with coefficient 1. a)

log ! 𝑥 + log ! 𝑦 !

b)

5 log ! 𝑥 + 2 log ! 𝑦 − log ! 𝑥𝑦

c)

6 log 𝑥 − 3 log 𝑦 !

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Chapter 8 – Properties of Log and Natural Log 5.

Solve each of the following equations for 𝑏. a)

6.

Practice Problems

b)

log ! 64 = 2

log ! 81 = 4

Solve each equation carefully. Be sure to check for extraneous solutions. Check your answer using your calculator. a)

log ! 𝑥 + 4 + log ! 𝑥 + 5 = 1

b)

log ! 6𝑥 ! − log ! 2𝑥 = 2

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Chapter 8 – Properties of Log and Natural Log

Practice Problems

8.2 – The Natural Logarithm 7.

8.

Solve each of the following equations. Give your answer in EXACT form and as an approximation rounded to three decimal places. a)

𝑒 !!!!! = 1

b)

42𝑒 !! = 100

c)

5 ln 𝑥 − 1.2 = 6.7

d)

ln 2𝑥 + 1 = 4

b)

𝑒 !" !

Simplify each of the following. a)

ln 𝑒 !

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Chapter 8 – Properties of Log and Natural Log 9.

Practice Problems

An advertising agency’s annual revenue has increased by 13% each year continually since the year 2010. The agency’s revenue can be modeled by 𝑅 𝑡 = 33𝑒 !.!"! , where 𝑡 is the number of years since 2010. a) What does the number 33 mean in the context of the problem?

b) What is the predicted revenue for 2016?

c) After how many years would the predicted revenue be $100 million?

d) In what year would the predicted revenue be $100 million?

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Chapter 8 – Properties of Log and Natural Log

Practice Problems

10. The population of the world can be modeled by 𝑃 𝑡 = 1.2𝑒 !.!"#! , in billions of people, where 𝑡 is the number of years since 1900. a) What was the predicted population of the world in 1985?

b) In which year would the predicted population be 10 billion people?

11. The percentage of Carbon-14 remaining in an object containing organic matter 𝑡 years since its death is modeled by 𝐴 𝑡 = 100𝑒 !!.!!!"#"! . a) What percentage of Carbon-14 would remain in an animal’s remains 200 years after its death?

b) If an archeologist finds bones of an animal that contain 85% of the original Carbon-14, how many years ago was the animal alive?

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Name _____________________________________

Date ____________________

Chapter 8 Assessment 1.

Fully expand log !

!"#! !!

using the properties of logarithms. In your final answer, the

argument in each logarithmic term should be a single variable.

2.

Use the properties of logarithms to condense log ! 𝑚 − log ! 𝑛 + log ! 𝑚𝑛! .!Your final answer should be an expression that is a single logarithm with coefficient 1.

3.

Solve the following equation carefully. Be sure to check for extraneous solutions. Check your answer using your calculator.

log ! 𝑥 + 11 + log ! 𝑥 + 5 = 3

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Chapter 8 – Properties of Logarithms and Natural Logarithm 4.

Solve each equation. Give your answer in exact form and as an approximation rounded to 3 decimal places. a)

5.

Assessment

1...