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.
Example of exponential functions:
Example of functions that are not exponential functions:
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. 2. 3. 4.
Graphing Exponential Functions
Graphing exponential functions in the form y = abx 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 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.
1. y =4•2x − 1 – 3 2. y =
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 defined as the value that as n gets larger and larger. As n goes to ∞ the approximate value of e to nine decimal places is: e ≈ 2.718281827. The irrational number e approximately 2.72, is called the natural base. The function 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.
1. y = ex 2. y =
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.
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.
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. cosZ3. tanL 4. sinX
5. cosL 6. tanZ /
In 7-10, find the trigonometric ratio that corresponds to each value and the angle given, using the triangle at the right.
7. 8.9. 10. /
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
Logarithmic Form: y = logbx Exponential Form: by = x
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 4.
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 = logbx
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 y = log2x
Characteristics of graphs of Logarithmic Functions
· Have vertical asymptotes
· Domain is restricted by vertical asymptote, however, range is
To Graph a logarithmic Functions in the form y = alogb(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 = alogb(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):
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.
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. secZ3. cotL 4. cscX
5. secL 6. tanZ /
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.
7.) 8.)
9.) 10.)
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)
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.
1. log7(7x) 2. 3. 4.
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
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 Changing to the Natural Base
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 = 4.
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:
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.
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:
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.
1.) / 2.)Practice: In 3-6, Use the given trigonometric equation to find the indicated trigonometric ratio.
3.) / 4.)5.) / 6.)
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
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?
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.
More Trig Review
Remember to use SOH-CAH-TOA & HO-HA-A0 to find the six trigonometric ratios.
Practice: In 1-2, Use the given trigonometric equation to find the remaining five trigonometric equations.
1.) / 2.)Practice: In 3-6, Use the given trigonometric equation to find the indicated trigonometric ratio.
3.) / 4.)5.) / 6.)
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. 2. 3. 4.
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. 10. 11.