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Math Analysis
Chapter 3 Notes: Exponential and Logarithmic Functions
Day 21: Section 3-1 Exponential Functions
3-1: Exponential Functions
After completing section 3-1 you should be able to do the following:
1. Evaluate exponential functions
2. Graph exponential functions
3. Evaluate functions with base e
4. Use compound interest formulas
An exponential functions are functions whose equations contain a variable in the exponent.
Definition of the Exponential Function
The exponential function f with base b is defined by:
f ( x)  b x or y  b x
Where b is a positive constant other than 1 (b > 0 and b ≠ 0) and x is any real number.
Example of exponential functions:
f ( x)  3
x
base of 3
h( x)  10
e( x)  2
x
base of 10
x 1
1
k ( x)   
2
x 3
base of 2
base of 1/2
Example of functions that are not exponential functions:
a( x)  1x
The base of an exponential
function must be a positive
constant other than 1.
b ( x )   2 
x
The base of an exponential
function must be a positive.
c( x)  x4
Variable is the
base and not the
exponent.
d ( x)  x x
Variable is both the base and
the exponent.
To evaluate expressions with exponents with a calculator:
1. Enter base
2. find the button [^] or [yx] and push it
3. enter exponent
4. push the equal button and you should have your answer.
Practice: Approximate each number using a calculator. Round your answer to three decimal places.
1. 32.5
2. 4 6
3. e 4
4. 63.1
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Math Analysis Notes Chapter 3 Mr. Hayden
Graphing Exponential Functions
Graphing exponential functions in the form y = ab x for b > 1 where a is a real number and b is the base (b ≠ 1)
Practice: (a) Graph each exponential function. (b) State the domain and range in interval notation. (c) Label the horizontal
asymptote.
1. y = 4x
x
y
2. y =
x
1 x
4
2
y
Graphing Exponential Functions in the form y = abx − h + k
 You must graph the parent function 1st. y = abx
 Then translate the graph horizontally according to h and vertically according to k.
 You must show both the parent function and the translated function in order to get credit when graphing exponential
functions that have translations in them.
Practice: (a) Graph each exponential function. (b) State the domain and range in interval notation. (c) Label the horizontal
asymptote.
2. y = 2 x 3  1
1. y =4•2x − 1 – 3
x
y
x
y
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Math Analysis Notes Chapter 3 Mr. Hayden
The Natural Base e
An irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. The number e is
n
 1
defined as the value that  1   as n gets larger and larger. As n goes to ∞ the approximate value of e to nine decimal places
 n
is:
e ≈ 2.718281827.
The irrational number e approximately 2.72, is called the natural base. The function
x
f ( x )  e is called the natural exponential function.
Practice: (a) Graph each exponential function. (b) State the domain and range in interval notation. (c) Label the horizontal
asymptote.
2. y = e x  3
1. y = ex
x
y
x
y
Formulas for Compound Interest
After t years, the balance A, in an account with principal P and annual interest rate r (in decimal form) is given by the
following formulas:
1.
 r
For n compoundings per year: A  P 1  
 n
2.
For continuous compounding: A  Pert
nt
Practice: A sum of $10,000 is invested at an annual rate of 8%. Find the balance in the account after 5
years subject to (a) quarterly compounding and (b) continuous compounding.
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Math Analysis Notes Chapter 3 Mr. Hayden
Geometry Review: Trigonometry
A ratio of the lengths of two sides of a right triangle is called a trigonometric ratio. The three most
common ratios are sine, cosine, and tangent. These three rations are defined for the acute angles of right
triangles, though your calculator will give you values of sine, cosine, and tangent for angles of greater
measure. The abbreviations for the ratios are sin, cos, and tan respectively.
sin A 
leg opposite to A opp a


hypotenuse
hyp c
cos A 
leg adjacent to A adj b


hypotenuse
hyp c
tan A 
leg opposite to A opp a


leg adjacent to A adj b
B
c
a
A
b
C
In 1-6, find the indicated trigonometric ratio using the right triangles to the right. Final answers should be in
reduced fractional form.
1. sinM
2. cosZ
Y
K
3. tanL
4
4. sinX
6 3
6
5. cosL
6. tanZ
Z
X
4 2
L
M
In 7-10, find the trigonometric ratio that corresponds to each value and the angle given, using the triangle at
the right.
7.
9
, G
41
8.
40
, H
9
H
41
9
9.
40
, H
41
10.
9
, H
41
E
40
G
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Math Analysis Notes Chapter 3 Mr. Hayden
Day 23: Section 3-2 Logarithmic Functions
3-2: Logarithmic Functions
After completing section 3-2 you should be able to do the following:
1. Change from logarithmic to exponential form.
2. Change from exponential to logarithmic form.
3. Evaluate logarithms.
4. Use basic logarithmic properties.
5. Graph logarithmic functions
6. Find the domain and range of a logarithmic function.
7. Use of common logarithms
8. Use natural logarithms
Definition of the Logarithmic Function
For x > 0 and b > 0, b ≠ 1:
y  log b x is equivalent to b y  x
The function f ( x)  logb x is the logarithmic function with base b.
The point of logarithmic functions is they allow us to solve for the value of a variable that is an exponent.
Exponent
Exponent
Exponential Form: by = x
Logarithmic Form: y = logbx
Base
Base
To change from logarithmic form to the more familiar exponential form, use this pattern:
y = logbx
means
by = x
Practice: In 1-4, Write each equation in its equivalent exponential form:
1. 3 = log7x
2. 2 = logb25
3. log426 = y
4. log28 = x
Practice: In 1-4, Write each equation in its equivalent logarithmic form:
1. 25 = 32
2. b3 = 27
3. ey = 33
4. 4x = 64
To Evaluate a logarithmic expression without using a calculator:
1. Set logarithmic expression equal to x
2. Write the equation in its equivalent exponential form
3. Evaluate the exponential expression
4. The answer to the exponential expression it the value of the logarithmic expression.
Practice: In 1-4, Evaluate each expression without using a calculator.
1. log416
2. log648
3. log264
1
4. log 4  
 16 
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Math Analysis Notes Chapter 3 Mr. Hayden
Basic Logarithmic Properties Involving One
1. logb b  1 because 1 is the exponent to which b must be raised to obtain b.
logb 1  0 because 0 is the exponent to which b must be raised to obtain 1. (b0 = 1)
2.
Inverse Properties of Logarithms
For b > 0 and b ≠ 1:
log b b x  x
The logarithm with base b of b raised
to a power equal that power
blogb x  x
b raised to the logarithm with base b
of a number equals that number
Practice: In 1-4 Evaluate each expression without using a calculator.
1. log99
2. log41
3. log773
4. 8log819
To Graph a logarithmic Functions in the form y = log bx
1. Rewrite the function in exponential form
2. Graph the exponential equation by making an x/y table. You will be choosing values for the exponent (in this case y)
3. Connect points with a smooth curve.
Practice: Graph y = 2x and y = log2x on the same rectangular coordinate system.
y = 2x
x
y = log2x
y
x
y
Characteristics of graphs of Logarithmic Functions
 Have vertical asymptotes
 Domain is restricted by vertical asymptote, however, range is  ,  
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Math Analysis Notes Chapter 3 Mr. Hayden
To Graph a logarithmic Functions in the form y = alog b(x – h) + k
1. Write logarithmic function that does not contain transformations h or k.
2. Write the logarithmic function found in step 1 in exponential form.
3. Follow steps above to graph the exponential function found in step 2.
4. Now use h to translate each point horizontally h-units and k to translate each point vertically k-units
5. Connect these new points with a smooth curve to get the graph of y = alog b(x – h) + k
Practice: (a) graph: y = 3 – 2log3(x – 1). (b) State the domain and range. (c) Write the equation of the asymptote of the
graph.
Practice: (a) graph: y = ln(x + 1) – 3. (b) State the domain and range. (c) Write the equation of the asymptote of the graph.
The common base (10):
The natural base (e):
A logarithm with a base of 10 is written
without a base. So log15 is read as “log base
10 of 15” or “the common log of 15”
A logarithm with a base of e is written a
natural logarithm. So loge15 is written as
ln15 and is read as “natural log of 15”
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Math Analysis Notes Chapter 3 Mr. Hayden
Practice: Evaluate or simplify each expression without using a calculator.
1. log100
2. lne
3. lne8
4. log104x
Geometry Review: Trigonometry
Besides the three most common trigonometric ratios, sine, cosine, and tangent, there are three more rations
that are considered the reciprocal ratios. These reciprocal ratios are cosecant, secant, and cotangent. The
abbreviations for the ratios are csc, sec, and cot respectively.
csc A 
hypotenuse
hyp c
1

 
leg opposite to A opp a sin A
sec A 
hypotenuse
hyp c
1

 
leg adjacent to A adj b cos A
cot A 
leg adjacent to A adj b
1

 
leg opposite to A opp a tan A
B
c
a
A
b
C
In 1-6, find the indicated trigonometric ratio using the right triangles to the right. Final answers should be in
reduced fractional form.
1. cscM
2. secZ
Y
K
3. cotL
4
4. cscX
6
5. secL
6. tanZ
6 3
Z
X
4 2
L
M
How to find another trigonometric equation given one trigonometric equation.
 Use the given information to draw a right triangle and making the given sides
 Find the missing side using Pythagorean Theorem
 Now that you have all there sides of the right triangle labeled you can write the trigonometric
equation for any ratio.
In 7-: Use the given trig equation to find the value of a different trig ratio.
4
5
7.) sin x  , cos x  ?
8.) tan x  , secx  ?
5
12
1
5
9.) cos x  , cscx  ?
10.) sin x  , cot x  ?
2
7
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Math Analysis Notes Chapter 3 Mr. Hayden
Day 24: Section 3-3 Properties of Logarithms; Section 3-4 Exponential and Logarithmic Equations
3-3: Properties of Logarithms
After completing section 3-3 you should be able to do the following:
1. Use the product rule
2. Use the quotient rule
3. Use the power rule
4. Expand logarithmic expressions
5. Condense logarithmic expressions
6. Use the change-or-base property
Rules of Logarithms (very similar to the rules of exponents)
Let b, M, and N be positive real numbers with b ≠ 1.
The Product Rule
logb(MN) = logbM + logbN
 The logarithm of a product is the sum of the logarithms.
The Quotient Rule
M 
logg    log b M  log b N
N
 The logarithm of a quotient is the difference of the logarithms.
The Power Rule
log b M x  xlog b M

The logarithm of a number with an exponent is the product of the exponent and the logarithm of that number.
We use these rules in order to expand or condense a logarithmic expression
Practice: 1-4, use the properties of logarithms to expand each logarithmic expression as much as possible. Where possible,
evaluate the logarithmic expression without using a calculator.
 8x2 3 1  x 
 x 

1. log7(7x)
2. log 
3. log 100x
4. log 2 

2
 7  x  1 
 100 
Practice: 1-3, use the properties of logarithms to condense each logarithmic expression. Write the expression as a single
logarithm whose coefficient is 1.
1. log(2x – 5) – 3log3
2. 3(logx + logy) – 2(log(x + 1))
3. 4lnx + 7lny – 3lnz
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Math Analysis Notes Chapter 3 Mr. Hayden
Using Change of base to evaluate Logarithms
Calculators can only evaluate logarithms that have the common base (10) or the natural base (e). We can change any base of a
logarithm by using the change of base property:
Changing to the Common Base
logM
log b M 
logb
Changing to the Natural Base
lnM
log b M 
lnb
Practice: Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places.
1. log513
2. log1487.5
3. log0.112
4. logπ60
3-4: Exponential and Logarithmic Equations
After completing section 3-4 you should be able to do the following:
1. Use like bases to solve exponential equations
2. Use logarithms to solve exponential equations
3. Use the definition of a logarithm to solve logarithmic equations
4. Use the one-to-one property of logarithms to solve logarithmic equations.
Two methods to solving exponential equations:
Method 1: Expressing each side as a power of the same base.
Practice: In 1-4, Solve:
1. 53x – 6 = 125
2. 8x + 2 = 4x – 3
3. 5x =
1
25
4. 6
x 3
4
 6
Method 2: Using Natural Logarithms to Solve Exponential Equations
Since most exponential equations cannot be rewritten so that each side has the same base. Logarithms are extremely useful in
solving such equations.
Steps to solve exponential equations using natural logarithms
1. Isolate the exponential expression.
2. Take the natural logarithm on both sides of the equation.
3. Simplify using one of the following properties:
lnbx = xlnb
or
lnex = x
4. Solve for the variable.
Practice: In 1-4 Solve:
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Math Analysis Notes Chapter 3 Mr. Hayden
1. 5x = 134
2. 7e2x – 5 = 58
3. 32x – 1 = 7x + 1
4. e2x – 8ex + 7 = 0
Logarithmic Equations
Steps to solve logarithmic equations
1. Get logarithm on one side of the equation and make sure the coefficient is 1. If not use algebraic properties to move
constants or coefficients to the other side of the equal sign if necessary.
2. Use the properties of logarithms to write the expression as a single logarithm whose coefficient is 1. (Condense if
necessary)
3. Use the definition of a logarithm to rewrite the equation in exponential form:
logbM = c means bc = M
4. Solve for the variable
5. Check proposed solutions in the original equation. Include in the solution set only values for which M > 0
Practice: In 1-4 Solve:
1. log2(x – 4) = 3
2. 4ln(3x) = 8
3. logx + log(x – 3) = 1
4. ln( x  3)  ln(7 x  23)  ln( x  1)
More Trig Review
A ratio of the lengths of two sides of a right triangle is called a trigonometric ratio. The six trigonometric
ratios are defined for the acute angles of a right triangle as:
sin A 
leg opposite to A opp a


hypotenuse
hyp c
csc A 
hypotenuse
hyp c
1

 
leg opposite to A opp a sin A
cos A 
leg adjacent to A adj b


hypotenuse
hyp c
sec A 
hypotenuse
hyp c
1

 
leg adjacent to A adj b cos A
tan A 
leg opposite to A opp a


leg adjacent to A adj b
cot A 
leg adjacent to A adj b
1

 
leg opposite to A opp a tan A
B
c
a
A
b
C
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Math Analysis Notes Chapter 3 Mr. Hayden
A harmonic that can be used to remember the 1st three trigonometric ratios: sine, cosine, and tangent
is SOH-CAH-TOA. To remember the reciprocal functions cosecant, secant, and cotangent you can
use “HO”, “HA” and “AO” respectively.
Practice: In 1-2, Use the given trigonometric equation to find the remaining five trigonometric
equations.
6
3
1.) sin X 
2.) csc G 
2
4
Practice: In 3-6, Use the given trigonometric equation to find the indicated trigonometric ratio.
3.) sec X  2, cot X  ?
2
, tan X  ?
4.) sin X 
2
5.) csc X  5, cos X  ?
3
6.) tan X  , sin X  ?
2
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Math Analysis Notes Chapter 3 Mr. Hayden
Day 25: Section 3-5 Exponential Growth and Decay; Modeling Data; Compound Interest Problems
3-5: Exponential Growth and Decay
After completing section 3-5 you should be able to do the following:
1. Model exponential growth and decay
2. Use compound interest formulas to solve word problems
Exponential Growth and Decay Models
The mathematical model for exponential growth or decay is given by:
A = a0ekt
Where a0 = original amount, or size of the growing or decaying entity at t = 0. A is the amount at time t. and k is a constant
representing the growth rate (many times given as a percentage).


If k is positive the function models a growth
If k is negative the function models a decay
Exponential Growth
k>0
Exponential Decay
k<0
Practice: The exponential model A = 106.2e.018t describes the population of a country, A, in millions, t years after 2003. Use this
model to solve Exercises 1-4.
1. What was the population of the country in 2003?
2. Is this county’s having a population growth or decay?
3. What will be the population in 2012?
4. When will the population be 1000 million?
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Math Analysis Notes Chapter 3 Mr. Hayden
Formulas for Compound Interest
After t years, the balance A, in an account with principal P and annual interest rate r (in decimal form) is given by the
following formulas:

 r
For n compoundings per year: A  P 1  
 n
nt
 For continuous compounding: A  Pert
Practice:
1. Find the number of years it takes for $10,000 to double at an interest rate of 7% compounded quarterly.
2. Find the number of years it takes $1500 to become $4000 at an interest rate of 5.5% compounded continuously.
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Math Analysis Notes Chapter 3 Mr. Hayden
More Trig Review
Remember to use SOH-CAH-TOA & HO-HA-A0 to find the six trigonometric ratios.
Hypotenuse
A
Si de
Op posi te
to A
Si de
Adj acent
to A
sin A 
opp
1

( SOH )
hyp csc A
csc A 
hyp
1

( HO)
opp sin A
cos A 
adj
1

(CAH )
hyp sec A
sec A 
hyp
1

( HA)
adj cos A
tan A 
opp
1

(TOA)
adj cot A
cot A 
adj
1

( AO)
opp tan A
Practice: In 1-2, Use the given trigonometric equation to find the remaining five trigonometric
equations.
2
5
1.) cos X 
2.) cot G 
6
3
Practice: In 3-6, Use the given trigonometric equation to find the indicated trigonometric ratio.
3.) csc X  3, tan X  ?
1
4.) sin K  , tan K  ?
2
5
5.) sec X  , sin X  ?
2
3
6.) cot X  , sec X  ?
2
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Math Analysis Notes Chapter 3 Mr. Hayden
Chapter 3 Review Sheet
Please complete each of the following problems on a separate sheet of paper. Show all of your work!
NO WORK = NO CREDIT!
For questions 1-4, graph each function by making a table of values. State the domain, range, and
equations of any asymptotes.
1. f ( x)  3 x  5
2. f ( x) 
f ( x)  log 1 x  4
1 x 4
2
2
3. f ( x)  log 2 ( x  2)
4.
3
For questions 5-8, solve each word problem.
5. Dustin deposits $1000 into his bank account at 4% annual interest rate. If the account is compounded
continuously, how long would it take for Dustin’s account to double?
6. Beth deposits $300 into her bank account at an interest rate of 7%. If the account is compounded weekly,
how long would it take for her account to triple?
7. Susan decides to save her money by putting it in a bank account that earns 3% annual interest. Susan puts
$2500 in an account whose interest is compounded quarterly. How much money is in Susan’s account after
8 years?
8. Chris deposits $50 into his account that earns 4% annual interest. If the account is compounded
continuously how long will it take for Chris to have $75 in his account?
For questions 9-14, solve each equation.
9. log 4 x  log 4 (2 x - 3)  log 4 2
10. 8  105 x 4  35
11. 34 x  813 x 2
12. log( x2  1)  log( x  5)
13. ln( x  3)  2  8
14.
2
log(2 x  4)  log( x  6)
For questions 15-16, expand each expression using your logarithmic properties.
15. log 3
81x 7 y 6
4
16. log 2 16 x8 y 2 ( z  2)
z
For questions 17-18, condense each expression into a single logarithm using your logarithmic
properties.
17. 4log5 x  2log5 y  3log5 z
18. 25log2 m  4log2 n  9log2 p  4log2 k
For questions 19-21, evaluate each expression without using your calculator.
1
19. log 1 256
20. log5 625
21. log 7
343
2
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Math Analysis Notes Chapter 3 Mr. Hayden