Exponential and Logarithm Natural Logs and e Exponential Growth and - - PDF document

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Exponential and Logarithm Functions

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Table of Contents

Exponential Functions Logarithmic Functions Properties of Logs Common Logs Natural Logs and e Exponential Growth and Decay

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Intro to Exponential Functions

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We have looked at linear growth, where the amount

• f change is constant.

X Y 1 3 2 5 3 7 4 9

If x = 5 what is y?

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When the rate of growth increases as time passes, the function is said to be exponential.

X Y 1 1 2 2 3 4 4 8

If x = 5 what is y?

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We will also looking at exponential decay. Think of it as you 8 m&m's and each day you eat half. How many will be left of the 5th day?

X Y 8 1 4 2 2 3 1 4 0.5

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From a Graph

The exponential function has a curved shape to it. For exponential growth, the function starts with the x-axis as an asymptote and increases to infinity. For exponential decay, the function starts at infinity and decreases to the x-axis as an asymptote. Exponential Growth Exponential Decay

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1 Which of these are exponential growth graphs?

A B C D E F G H I

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2 Which of these are exponential decay graphs?

A B C D E F G H I

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The general form of an exponential is

where x is the variable and a, b, and c are constants.

b is the growth rate.

If b > 1 then its exponential growth If 0< b < 1 then its exponential decay

c is the horizontal asymptote a + c is the y-intercept

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3 Consider the following equation is it exponential growth

• r decay?

A

growth

B

decay

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4 Considering the following equation, what is the equation of the horizontal asymptote?

A

y=2

B

y=3

C

y=4

D

y=5

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5 Considering the following equation, what is the equation

• f the y-intercept?

A

(0,2)

B

(0,3)

C

(0,4)

D

(0,5)

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6 Consider the following equation is it exponential growth

• r decay?

A

growth

B

decay

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7 Considering the following equation, what is the equation of the horizontal asymptote?

A

y=.2

B

y=1

C

y=3

D

y=4

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8 Considering the following equation, what is the equation

• f the y-intercept?

A

(0,.2)

B

(0,1)

C

(0,3)

D

(0,4)

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9 Consider the following equation is it exponential growth

• r decay?

A

growth

B

decay

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10 Considering the following equation, what is the equation of the horizontal asymptote?

A

y=0

B

y=1

C

y=3

D

y=4

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11 Considering the following equation, what is the equation

• f the y-intercept?

A

(0,0)

B

(0,1)

C

(0,3)

D

(0,4)

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Sketching the graph of an exponential requires using a, b, and c. 1) Identify horizontal asymptote (y = c) 2) Determine if graph is decay or growth 3) Graph y-intercept (0,a+c) 4) Sketch graph Example: Step 1 Step 2 Step 3 Step 4

y = 2

growth

(0,5)

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Graph

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Logarithmic Functions

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Logarithm functions are the inverses of exponential functions. Logs have the same domain as the exponential had range, that is a you cannot take the log of 0 or a negative. Exponential Log

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Logarithms are a way to rewrite an exponential equation. Rewrite the following in logarithmic form.

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Rewrite the following in exponential form.

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12 Which of the following is the correct logarithmic form of ?

A B C D

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13 Which of the following is the correct logarithmic form of ?

A B C D

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14 Which of the following is the correct exponential form of ?

A B C D

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15 Which of the following is the correct exponential form of ?

A B C D

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One way to solve a log equation (or expression) is to convert it back exponential form. Examples:

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16 Solve

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17 Solve

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18 Evaluate

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19 Solve

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Properties of Logs

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Properties of Logs

These rules may seem strange but recall that logs are a way of dealing with exponents, so when we multiplied like bases we added the exponents. Just like rule 1.

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Examples: Use the Properties of Logs to expand

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20 Which choice is the expanded form of the following

A B C D

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21 Which choice is the expanded form of the following

A B C D

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22 Which choice is the expanded form of the following

A B C D

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23 Which choice is the expanded form of the following

A B C D

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Use and to approximate the value

• f the expression.

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24 Approximate the value of the expression given and .

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25 Approximate the value of the expression given and .

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26 Approximate the value of the expression given and .

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Examples: Use the Properties of Logs to rewrite as a single log.

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27 Which choice is the contracted form of the following

A B C D

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28 Which choice is the contracted form of the following

A B C D

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29 Which choice is the contracted form of the following

A B C D

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30 Which choice is the contracted form of the following

A B C D

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Solving Log Equations Revisited

Equations will not always be in

• r
• form. We will need to use the

properties to convert them to one these forms.

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Extraneous answers: You can not take a log of x<0.

Check to see if your solution makes the value positive. r =3 works for each of the logs, but r= -1 is extraneous.

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31 Solve the following equation:

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32 Solve the following equation:

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33 Solve the following equation:

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34 Solve the following equation:

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Common Logs

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Common logs are logs with bases of 10. Notice the common log is written with an understood base of 10. Common logs are used in the Richter Scale to Decibels. Values for common logs can be calculated from a chart

• r by using a scientific calculator.

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Solving Exponential Equations Using Common Logs

We can solve the variable as exponent using common logs Recall We can use this method to introduce common logs and then use are preferred method to find the value of the common log.

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35 Solve the following equation.

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36 Solve the following equation.

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37 Solve the following equation.

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38 Solve the following equation.

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Logarithmic functions can be rewritten as the ratio of common logs. The advantage of this is that we have a method of calculating the value of common logs. Find the approximate value of

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39 Find the approximate value of

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40 Find the approximate value of

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41 Find the approximate value of

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Base e and Natural Logs

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e has a constant value of about 2.71828

e is the number used when something is continually growing, like a bacteria colony or an oil spill. The Natural Log is the inverse function of a base e function.

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Work with e and ln the same way you did 10 and log. For example:

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Write the following in the equivalent exponential or log form.

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Solve the following equations.

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Solve the following equations. Only 60.6 is a solution because ln(3(-60.6)) and ln(2(-60.6)) are undefined.

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The amount of money in a savings account, A, can be found using the continually compounded interest formula of A=Pert , where P is the principal (amount deposited), r is the annual interest rate (in decimal form), and t is time in years. If $500 is invested at 4% for 2 years, what will account balance be?

$541.64 in 2 years.

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Again,using the compounded continually formula of A=Pert If $500 is invested at 4% , how long until the account balance is doubled?

It will take about 17.3 years for the money to double.

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42 Find the value of x.

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43 Find the value of x.

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44 Find the value of x.

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45 Find the value of x.

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46 Find the value of x.

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47 Find the value of x.

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48 The amount of money in a savings account, A, can be found using the continually compounded interest formula of A=Pert , where P is the principal (amount deposited), r is the annual interest rate (in decimal form), and t is time in years. If $1000 is invested at 4% for 3 years,what is the account balance?

HINT Slide 84 / 99

49 The amount of money in a savings account, A, can be found using the continually compounded interest formula

• f A=Pert ,

where P is the principal (amount deposited), r is the annual interest rate (in decimal form), and t is time in years. If $1000 is invested at 4%, how long until the account balance is doubled?

HINT

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Applied Exponential Growth and Decay

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Formulas to remember: Simple Interest Compounded Interest(annually) Compounded Interest Compounded Continually (instantaneously)

Abbreviations I = interest P= principal (deposit) r= interest rate (decimal) t= time n= number of compoundings in one unit of t

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Examples: A bacteria constantly grows at a rate of 10% per hour, if initially there were 100 how long till there were 1000? It will take the colony 23 hours to increase its population from 100 to 1000.

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A new car depreciates in value at a rate of 8% per year. If a 5 year old car is worth $20,000,how much was it

• riginally worth?

Since the value of the car is going down, the rate is -.08 The original value of the car was about $30,345.

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The local bank pays 4% monthly on its savings account, how long would it take for a deposit, left untouched, to double?

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A certain radioactive material has a half-life of 20 years. If 100g were present to start, how much will remain in 7 years? Use half-life of 20 years to find r. In 7 years, about 78.5g remain.

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50 If you need your money to double in 8 years, what must the interest rate be if is compounded continually?

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51 If you need your money to double in 8 years, what must the interest rate be if is compounded annually?

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52 If you need your money to double in 8 years, what must the interest rate be if is compounded quarterly?

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53 If an oil spill widens continually at a rate of 15% per hour, how long will it take to go from 2 miles wide to 3miles wide?

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54 NASA calculates that a communications satellite's orbit is decaying exponentially at a rate of 12% per day. If the satellite is 20,000 miles above the Earth. How long till it is visible to the naked eye at 50 miles high, assuming it doesn't burn up on reentry?

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55 If the half-life of an element is 50 years, at what rate does it decay?

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56 If the half-life of an element is 50 years, how much of the element is left in 10 years?

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57 If the half-life of an element is 50 years, how much of the element is left in 15 years?

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58 If the half-life of an element is 50 years, how much of the element is lost between years 10 and 15?