Application of Logarithms

Click on the title below you wish to view Table of Contents Click on the title below you wish to view At the end of the presentation you will find a link to return to the Table of Contents I. Compound Interest Formula II. Continuous Compound Interest Formula III. Population Growth Formula IV. Doubling-Time Growth Formula V. Exponential Decay VI. Half-Life Decay Formula VII. Newton’s Law of Cooling

Compound Interest Formula

P is the Principle invested The amount of money invested in a Savings Institution The value of an object: House New Car Piece of Property Bicycle

total value of the investment after t years A is the total value of the investment after t years

number of years the Principle has been invested t is the number of years the Principle has been invested

r is the Rate of interest Expressed as a decimal

Number of times the interest is n is the Number of times the interest is Compounded per year “Compounded annually” n = 1

Number of times the interest is n is the Number of times the interest is Compounded per year “Compounded semi-annually” n = 2

Number of times the interest is n is the Number of times the interest is Compounded per year “Compounded quarterly” n = 4

Number of times the interest is n is the Number of times the interest is Compounded per year “Compounded monthly” n = 12

Example

One thousand dollars is invested at 12% interest compounded annually. Determine how much the investment is worth after 2 years.

One thousand dollars is invested at 12% interest compounded annually. Determine how much the investment is worth after 2 years.

One thousand dollars is invested at 12% interest compounded annually. Determine how much the investment is worth after 2 years.

One thousand dollars is invested at 12% interest compounded annually. Determine how much the investment is worth after 2 years.

One thousand dollars is invested at 12% interest compounded annually. Determine how much the investment is worth after 2 years.

One thousand dollars is invested at 12% interest compounded annually. Determine how much the investment is worth after 2 years.

One thousand dollars is invested at 12% interest compounded annually. Determine how much the investment is worth after 2 years.

One thousand dollars is invested at 12% interest compounded annually. Determine how much the investment is worth after 2 years.

One thousand dollars is invested at 12% interest compounded annually. Determine how much the investment is worth after 2 years.

One thousand dollars is invested at 12% interest compounded annually. Determine how much the investment is worth after 2 years.

One thousand dollars is invested at 12% interest compounded annually. Determine how much the investment is worth after 2 years.

Substitute

Calculator Ready Form

One thousand dollars is invested at 12% interest compounded annually. Determine how much the investment is worth after 2 years. The investment is worth $1,254.40

Next Example

The value of a new $500 television Decreases 10% per year. Find its value after 5 years.

The value of a new $500 television Decreases 10% per year. Find its value after 5 years. You Determine

The value of a new $500 television Decreases 10% per year. Find its value after 5 years.

The value of a new $500 television Decreases 10% per year. Find its value after 5 years.

The value of a new $500 television Decreases 10% per year. Find its value after 5 years.

The value of a new $500 television Decreases 10% per year. Find its value after 5 years.

The value of a new $500 television Decreases 10% per year. Find its value after 5 years.

Substitute

Calculator Ready Form

The television is worth $295.25

Next Example

One hundred dollars is invested at 7.2% interest compounded quarterly. How long will it take for the investment to double?

You Determine One hundred dollars is invested at 7.2% interest compounded quarterly. How long will it take for the investment to double? You Determine

One hundred dollars is invested at 7.2% interest compounded quarterly. How long will it take for the investment to double?

One hundred dollars is invested at 7.2% interest compounded quarterly. How long will it take for the investment to double?

One hundred dollars is invested at 7.2% interest compounded quarterly. How long will it take for the investment to double?

One hundred dollars is invested at 7.2% interest compounded quarterly. How long will it take for the investment to double?

t = One hundred dollars is invested at 7.2% interest compounded quarterly. How long will it take for the investment to double? unknown t =

t = One hundred dollars is invested at 7.2% interest compounded quarterly. How long will it take for the investment to double? unknown t =

Substitute t = unknown

t = unknown

Exponential Expression Isolate I s o l a t e Exponential Expression

LOG IT

Calculator Ready Form

It takes approximately 10 years Nearest Year Link Back to Table of Contents It takes approximately 10 years

Continuous Compound Interest Formula

Where did this formula come from?

If Compounded Continually Then how many times per year? Then n is approaching infinity

n approaches infinity r remains constant x approaches infinity Then Reciprocal n approaches infinity r remains constant Then x approaches infinity

Reciprocal of Both Sides

Study

x approaches infinity

x approaches infinity 2nd TABLE

x approaches infinity x 4 8 12 100 365 10,000 100,000 1,000,000 2.4414 2.5658 2.6130 2.7048 2.7146 2.7182 2.7183

x approaches infinity x 4 8 12 100 365 10,000 100,000 1,000,000 2.4414

x approaches infinity 2.4414 2.5658 x 4 8 12 100 365 10,000 100,000 1,000,000 2.4414 2.5658

x approaches infinity 2.4414 2.5658 2.6130 x 4 8 12 100 365 10,000 100,000 1,000,000 2.4414 2.5658 2.6130

x approaches infinity 2.4414 2.5658 2.6130 2.7048 x 4 8 12 100 365 10,000 100,000 1,000,000 2.4414 2.5658 2.6130 2.7048

x approaches infinity 2.4414 2.5658 2.6130 2.7048 2.7146 x 4 8 12 100 365 10,000 100,000 1,000,000 2.4414 2.5658 2.6130 2.7048 2.7146

x approaches infinity 2.4414 2.5658 2.6130 2.7048 2.7146 2.7182 x 4 8 12 100 365 10,000 100,000 1,000,000 2.4414 2.5658 2.6130 2.7048 2.7146 2.7182

x approaches infinity x 4 8 12 100 365 10,000 100,000 1,000,000 2.4414 2.5658 2.6130 2.7048 2.7146 2.7182 2.7183

x approaches infinity approaching 2.4414 2.5658 2.6130 2.7048 2.7146 12 100 365 10,000 100,000 1,000,000 2.4414 2.5658 2.6130 2.7048 2.7146 2.7182 2.7183 approaching

x approaches infinity

x approaches infinity

Continuous Compound Interest Formula

Next Example

Mrs. Johnson received a bonus equivalent to 10% of her yearly salary and has decided to deposit it in a savings account in which interest is compounded continuously. Her salary is $38,500 per year and the account pays 7.5% interest. How long will it take for her deposit to double in value?

Mrs. Johnson received a bonus equivalent to 10% of her yearly salary and has decided to deposit it in a savings account in which interest is compounded continuously. Her salary is $38,500 per year and the account pays 7.5% interest. How long will it take for her deposit to double in value?

Mrs. Johnson received a bonus equivalent to 10% of her yearly salary and has decided to deposit it in a savings account in which interest is compounded continuously. Her salary is $38,500 per year and the account pays 7.5% interest. How long will it take for her deposit to double in value?

Mrs. Johnson received a bonus equivalent to 10% of her yearly salary and has decided to deposit it in a savings account in which interest is compounded continuously. Her salary is $38,500 per year and the account pays 7.5% interest. How long will it take for her deposit to double in value?

Mrs. Johnson received a bonus equivalent to 10% of her yearly salary and has decided to deposit it in a savings account in which interest is compounded continuously. Her salary is $38,500 per year and the account pays 7.5% interest. How long will it take for her deposit to double in value?

Mrs. Johnson received a bonus equivalent to 10% of her yearly salary and has decided to deposit it in a savings account in which interest is compounded continuously. Her salary is $38,500 per year and the account pays 7.5% interest. How long will it take for her deposit to double in value?

Mrs. Johnson received a bonus equivalent to 10% of her yearly salary and has decided to deposit it in a savings account in which interest is compounded continuously. Her salary is $38,500 per year and the account pays 7.5% interest. How long will it take for her deposit to double in value?

Substitute

Exponential Expression Isolate I s o l a t e Exponential Expression

1 To the nearest year Calculator Ready Form

1 To the nearest year

Next Example

Following the birth of child, a parent wants to make an initial investment of $2000 and would like it to grow to a value of $10,000 by the Childs 20th birthday. What annual interest rate compounded continually should the parent look for?

Following the birth of child, a parent wants to make an initial investment of $2000 and would like it to grow to a value of $10,000 by the Childs 20th birthday. What annual interest rate compounded continually should the parent look for?

Following the birth of child, a parent wants to make an initial investment of $2000 and would like it to grow to a value of $10,000 by the Childs 20th birthday. What annual interest rate compounded continually should the parent look for? ?

Following the birth of child, a parent wants to make an initial investment of $2000 and would like it to grow to a value of $10,000 by the Childs 20th birthday. What annual interest rate compounded continually should the parent look for? ?

Following the birth of child, a parent wants to make an initial investment of $2000 and would like it to grow to a value of $10,000 by the Childs 20th birthday. What annual interest rate compounded continually should the parent look for? ?

Following the birth of child, a parent wants to make an initial investment of $2000 and would like it to grow to a value of $10,000 by the Childs 20th birthday. What annual interest rate compounded continually should the parent look for? ?

Following the birth of child, a parent wants to make an initial investment of $2000 and would like it to grow to a value of $10,000 by the Childs 20th birthday. What annual interest rate compounded continually should the parent look for?

Substitute

Exponential Expression Isolate I s o l a t e Exponential Expression

1

Nearest hundredth of a percent Calculator Ready Form

= 0.0805 8.05%

Next Example

How long will it take to double and investment at 4.5% interest Doubling Time How long will it take to double and investment at 4.5% interest

How long will it take to double and investment at 4.5% interest

How long will it take to double and investment at 4.5% interest

How long will it take to double and investment at 4.5% interest

How long will it take to double and investment at 4.5% interest

How long will it take to double and investment at 4.5% interest

How long will it take to double and investment at 4.5% interest

How long will it take to double and investment at 4.5% interest Substitute

How long will it take to double and investment at 4.5% interest Solve for t to the nearest hundredth

1

How long will it take to double and investment at 4.5% interest Solve for t to the nearest hundredth

Next Example

You solve for r to the nearest tenth of a percent What interest rate should an investor seek to double his money in 20 years? You solve for r to the nearest tenth of a percent You Try Using

You solve for r to the nearest tenth of a percent What interest rate should an investor seek to double his money in 20 years? You solve for r to the nearest tenth of a percent

You Try

Will invests $2000 in a bond trust that pays 9% interest compounded semiannually. His friend Henry invests $2000 in a Certificate of Deposit that pays 8 ½ % compounded continuously. Who will have more money after 20 years, Will or Henry? How much more money?

Will invests $2000 in a bond trust that pays 9% interest compounded semiannually. His friend Henry invests $2000 in a Certificate of Deposit that pays 8 ½ % compounded continuously. Who will have more money after 20 years, Will or Henry? How much more money?

Will $684.84

You Try

Pat, who is 35, also invests $2000 in an IRA. At the age of 25 Coris invests $2000 in an Individual Retirement Account (IRA) that is allowed to accumulate interest tax-free until she retires. Pat, who is 35, also invests $2000 in an IRA. Pat and Coris each earn 8% annually compounded continuously, and each withdraws the funds from the account at age 65. To the nearest hundred, how much more money does Coris collect than Pat?

At the age of 25 Coris invests $2000 in an Individual Retirement Account (IRA) that is allowed to accumulate interest tax-free until she retires. Pat, who is 35, also invests $2000 in an IRA. Pat and Coris each earn 8% annually compounded continuously, and each withdraws the funds from the account at age 65. To the nearest hundred, how much more money does Coris collect than Pat? Link Back to Table of Contents $27,000

Population Growth

is the size of the original population

P is the size of the population after t years

r annual growth rate of the population

t is the number of years

Example

The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987 a) Using these statistics, find the average growth rate of the population in Raleigh-Durham (to the nearest tenth of a percent) and determine an equation of the population growth curve for this region. b) Assuming a constant growth rate, predict the population for Raleigh-Durham in 1995 to the nearest unit.

The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987 a) Using these statistics, find the average growth rate of the population in Raleigh-Durham (to the nearest tenth of a percent) and determine an equation of the population growth curve for this region. b) Assuming a constant growth rate, predict the population for Raleigh-Durham in 1995 to the nearest unit. Find r

The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987

The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987

The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987

The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987

The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987 unknown 560,774 665,400

The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987 unknown 560,774 7 665,400

Find the population growth rate for Raleigh-Durham, North Carolina to the nearest tenth of a percent unknown 560,774 7 665,400

unknown 560,774 7 665,400

OR Law 2

Nearest tenth of a percent Calculator Ready Form Nearest tenth of a percent

Next Step

We need to find an equation that MODELS the population growth for Raleigh-Durham, North Carolina You Find

This is the equation that MODELS the population growth in Raleigh-Durham, North Carolina

Next Step

Use this equation to project the population in Raleigh-Durham, North Carolina in 1995 to the nearest unit

t To find P we need to know the value of ? Use this equation to project the population in Raleigh-Durham, North Carolina in 1995 to the nearest unit To find P we need to know the value of ? t

Use this equation to project the population in Raleigh-Durham, North Carolina in 1995 to the nearest unit The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987 t = 15

Use this equation to project the population in Raleigh-Durham, North Carolina in 1995 to the nearest unit

Link Back to Table of Contents Use this equation to project the population in Raleigh-Durham, North Carolina in 1995 to the nearest unit Link Back to Table of Contents

Doubling-Time Growth Formula

is the size of the Original Population Bacteria Culture of Yeast ETC.

Population after t time is the size of the Population after t time Years Weeks Minutes Months Hours Seconds ETC.

is the amount of time it takes for the population to double Years Weeks Minutes Months Hours Seconds ETC.

is the time period Years Weeks Minutes Months Hours Seconds ETC.

A certain bacteria population doubles in size every 12 hours A certain bacteria population doubles in size every 12 hours. By how much will it grow in 2 days?

A certain bacteria population doubles in size every 12 hours A certain bacteria population doubles in size every 12 hours. By how much will it grow in 2 days?

A certain bacteria population doubles in size every 12 hours A certain bacteria population doubles in size every 12 hours. By how much will it grow in 2 days?

A certain bacteria population doubles in size every 12 hours A certain bacteria population doubles in size every 12 hours. By how much will it grow in 2 days?

A certain bacteria population doubles in size every 12 hours A certain bacteria population doubles in size every 12 hours. By how much will it grow in 2 days?

A certain bacteria population doubles in size every 12 hours A certain bacteria population doubles in size every 12 hours. By how much will it grow in 2 days?

A certain bacteria population doubles in size every 12 hours A certain bacteria population doubles in size every 12 hours. By how much will it grow in 2 days?

Substitute

The population grows by a factor of 16 in 2 days Link Back to Table of Contents The population is 16 times greater than the size of the original population The population grows by a factor of 16 in 2 days

EXPONENTIAL DECAY

Graphs

Exponential Growth Exponential Decay

Exponential Growth Exponential Decay

Exponential Growth Exponential Decay

Exponential Growth Exponential Decay

Exponential Growth Exponential Decay

General Form Exponential Equation a is a constant Exponential Growth Exponential Decay

You Try

You may want check your answers with your graphing calculator Determine whether each equation represents an exponential growth or an exponential decay curve. You may want check your answers with your graphing calculator

Exponential Growth Exponential Decay Exponential Growth Determine whether each equation represents an exponential growth or an exponential decay curve. You may want check your answers with your graphing calculator Exponential Growth Link Back to Table of Contents Exponential Decay Exponential Growth Exponential Growth

Half-Life Decay Formula

Radioactive Substances Used for Radioactive Substances

Rewrite

is the original quantity of the radioactive substance (isotope)

A is the amount of radioactive substance after t years Smaller Larger or Smaller Amount?

of the radioactive substance h is the half-life of the radioactive substance

t is the number of years

The half life of radioactive radium (226Ra) is 1599 years. Given an initial quantity of 10 grams, how much will remain after 1000 years.

The half life of radioactive radium (226Ra) is 1599 years The half life of radioactive radium (226Ra) is 1599 years. Given an initial quantity of 10 grams, how much will remain after 1000 years. You Find

You Finish & Round to the Nearest Hundredth The half life of radioactive radium (226Ra) is 1599 years. Given an initial quantity of 10 grams, how much will remain after 1000 years. You Finish & Round to the Nearest Hundredth

Link Back to Table of Contents grams Link Back to Table of Contents

Newton's Law of Cooling

is the Original temperature of the object

is the temperature of the surrounding air

is the final temperature of the object after t minutes

is the rate at which the object is cooling

When the soup is removed from the pot, it is 212ºF. A chef wants the soup to be served to the customer at a temperature of no less than 160º F. It has been determined that the cooling rate r for this soup is 0.21ºF per minute. When the soup is removed from the pot, it is 212ºF. If the room temperature in the restaurant is 68ºF, determine in how many minutes the soup must be served to meet the chef’s request. To the nearest minute

When the soup is removed from the pot, it is 212ºF. A chef wants the soup to be served to the customer at a temperature of no less than 160º F. It has been determined that the cooling rate r for this soup is 0.21ºF per minute. When the soup is removed from the pot, it is 212ºF. If the room temperature in the restaurant is 68ºF, determine in how many minutes the soup must be served to meet the chef’s request.

When the soup is removed from the pot, it is 212ºF. A chef wants the soup to be served to the customer at a temperature of no less than 160º F. It has been determined that the cooling rate r for this soup is 0.21ºF per minute. When the soup is removed from the pot, it is 212ºF. If the room temperature in the restaurant is 68ºF, determine in how many minutes the soup must be served to meet the chef’s request.

When the soup is removed from the pot, it is 212ºF. A chef wants the soup to be served to the customer at a temperature of no less than 160º F. It has been determined that the cooling rate r for this soup is 0.21ºF per minute. When the soup is removed from the pot, it is 212ºF. If the room temperature in the restaurant is 68ºF, determine in how many minutes the soup must be served to meet the chef’s request.

When the soup is removed from the pot, it is 212ºF. A chef wants the soup to be served to the customer at a temperature of no less than 160º F. It has been determined that the cooling rate r for this soup is 0.21ºF per minute. When the soup is removed from the pot, it is 212ºF. If the room temperature in the restaurant is 68ºF, determine in how many minutes the soup must be served to meet the chef’s request.

When the soup is removed from the pot, it is 212ºF. A chef wants the soup to be served to the customer at a temperature of no less than 160º F. It has been determined that the cooling rate r for this soup is 0.21ºF per minute. When the soup is removed from the pot, it is 212ºF. If the room temperature in the restaurant is 68ºF, determine in how many minutes the soup must be served to meet the chef’s request.

Isolate

Law #2 Law #3

Law #2 Law #3 1 Calculator Ready Form To the nearest minute

Law #2 Law #3 1 Calculator Ready Form To the nearest minute 2 min

You Try

To the nearest hundredth of a minute Given that the original temperature of the coffee is 155ºF and the room temperature is 75ºF, determine after how many minutes the coffee will be 110ºF To the nearest hundredth of a minute

To the nearest hundredth of a minute Given that the original temperature of the coffee is 155ºF and the room temperature is 75ºF, determine after how many minutes the coffee will be 110ºF To the nearest hundredth of a minute Link Back to Table of Contents 2.75 min