10'1 Real Exponents and Exponential Functions

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10.1 Real Exponents and Exponential Functions

• Objective
• To simplify expressions and solve equations and
  inequalities involving real exponents

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• Exponential Function An exponential function y
  bx, where b gt 0, b ? 1, and x is a real
  numbers.
• y bx, where b gt 1 Also known as exponential
  ____________.
• a) As x approaches positive infinity, the
  function increases exponentially (very rapidly)
• b) As x approaches negative infinity, the
  function approaches the x-axis (the x-axis is an
  asymptote)
• c) f(0) 1
• 1)

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• Exponential Function An exponential function y
  bx, where b gt 0, b ? 1, and x is a real
  numbers.
• y bx, where 0 lt b lt 1 Also known as exponential
  __________
• a) As x approaches positive infinity, the
  function approaches the x-axis (the x-axis is an
  asymptote)
• b) As x approaches negative infinity, the
  function increases exponentially (very rapidly)
• c) f(0) 1
• 2)

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Two Condition of Exponential Functions
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Review of Exponent Rules from Chapter 5
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• Simplify. (Make sure the base is the same)
• 3) 4)

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• Simplify. (Make sure the base is the same)
• 5) 6)

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• Solve. (Get the same base if possible)
• 7) 8)

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• Solve. (Get the same base if possible)
• 9) 10)

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• Simplify. (Get the same base if possible)
• 11) 12)

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• Use calculator to evaluate and round to the
  tenths place.
• 13)

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• Assignment 10.1
• Page 600 (19-31 odd), (39-47 odd)
• Use a graphing calculator to graph 49-51. State
  the domain, range, intercepts, asymptotes,
  increasing or decreasing function. Also label 3
  points on your graph.
• 55(a-d), 58, 62, 64, 65, 66, 67

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10.2 Logarithms and Logarithmic Functions

• Objectives
• To write exponential equations in logarithmic
  form and vice versa
• To evaluate logarithmic expressions
• To solve equations and inequalities involving
  logarithmic functions

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• Write in log form. b base, a answer, e
  exponent
• logarithmic form exponential form
• 1) 2) 3)

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• Write in exponential form. b base, a answer,
  e exponent
• logarithmic form exponential form
• 4) 5) 6)

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• Evaluate each expression.
• 7) 8) 9)

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• Solve each equation.
• 10) 11) 12)

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• Solve each equation.
• 13) 14)

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• Solve each equation.
• 15) 16)

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• Solve each equation.
• 17)

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• Assignment 10.2
• Page 608 (6-8), (21-47 odd), 59, 60, 62, 63, 64,
  66, 67, 68, 69

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10.3 Properties of Logarithms

• Objective
• To simplify and evaluate expressions using
  properties of logarithms
• To solve equations involving logarithms

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• Decompose.
• 1) 2)

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• Recompose.
• 3) 4)

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• 5) 6)

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• 7) 8)

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• Solve each equation.
• 9) 10)

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• Solve each equation.
• 11) 12)

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• Solve each equation.
• 13) 14)

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• Solve using change of base.
• 15) 16)

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• Assignment 10.3
• Page 615 (17-43 odd), 47, 48, 49, 50, 51, 53, 54

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10.4 Common Logarithms

• Objective
• 1) To find common logarithms

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• Base 10 logarithms are called common logarithms
• log10x log x
• log 10 _____
• log 100 _____
• log 1000 _____

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• Use a calculator to find the log for each and
  round 4 decimal places.
• 1) 64.7 2) 0.047 3) 0.0035

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• Use a calculator to find the antilogarithm for
  each and round 4 decimal places.
• 4) 0.3142 5) -0.2615 6) 0.5734

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• Assignment 10.4
• Page 620 (19-29 odd) dont worry about stating
  the mantissa and characteristic
• 33, 34, 35, 37, 38,
• Self Test
• Page 621 (1, 3, 5, 7)

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10.5 Natural Logarithms

• Objective
• To find natural logarithms of numbers

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• e is an irrational number whose value is
  approximately 2.718
• Base e logarithms are called natural logarithms
• loge x ln x
• If x ey, then y loge x and y ln x
• ln e _____

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Use the compound interest formula A P(1
r/n)nt to complete the following chart when P
1, r 100 and t 1 year. The number of times
the investment is compounded annually is n.The
final amount is A.
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The Number e
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• The Number e
• The special irrational number you arrived at
  above, 2.718, is quite common and therefore it
  received its own name. e! It is used in
  computing problems that have continuous growth or
  decay.
• Below are two formulas that are often used for
  situations involving continuous growth or decay.
  They are really the same formula. They just have
  different variable names for the same things.
• The equation A Pert, where P is the initial
  amount, A is the final amount, r is the annual
  interest rate, and t is the time in years, is
  used for calculating interest that is compounded
  continuously.
• The equation y nekt, where y is the final
  amount, n is the initial amount, k is a constant
  of variation and t is time. This is used for
  continuous exponential growth or decay in
  situations involving nature.

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• Use a calculator to find each and round 4 decimal
  places.
• ln 2.68 2) ln 0.045 3) x if ln x 0.75

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• Use a calculator to find each and round 4 decimal
  places.
• 4) x if log x 5.4 5) antiln -2.003 6) antlog
  63.2

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• Assignment 10.5
• Page 624 (11-27 odd), 35a, 37, 38, 39, 43

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10.6 Solving Exponential Equations

• Objective
• To solve equations with variable exponents by
  using logarithms
• To evaluate expressions involving logarithms with
  different bases
• To solve problems by using estimation

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• Solve. Round 3 decimal places out.
• 1) 2)

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• Solve. Round 3 decimal places out.
• 3) 4)

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• Find the value of each logarithm to 3 decimal
  places.
• 5) 6) 7)
• (Use change of base formula )

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Natural Log Formulas(Base e logs)
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• 8) The population of Moreville, Montana is 5200.
  The growth rate is growing exponentially at
  3.25. Use the formula
• to figure out how long it will take for the
  town population to triple.

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• Assignment 10.6
• Page 629 (13-45 odd), 47, 48, 49, 51, 52
• 37 should be

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10.7 Growth and Decay

• Objective
• To use logarithms to solve problems involving
  growth and decay

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Natural Log Formulas(Base e logs)
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Common Logarithm Formulas(Base 10 logs)
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• 1) For a certain radioactive element, k is -0.377
  when t is measured in days. How long will it take
  500 grams of the element to reduce to 200 grams?

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• 2) The Jamesons bought a new house 5 years ago
  for 65,000. The house is now worth 117,000.
  Assuming a steady rate of growth, what was the
  yearly rate of appreciation?

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• 3) The town of Springfield grew from 324,270 in
  1990 to 458,440 in 2000. What was the rate of
  growth during this time? Use this rate of growth
  to predict the population in 2020.

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• 4) A certain element decays at a rate of
  -0.07654. Find the half-life of this element.

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• 5) A certain strain of bacteria grows from 40 to
  326 in 120 minutes. Find the k for the growth
  formula.

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• 6) A 40,000 car depreciates at a constant rate
  of 12 per year. In how many years will the car
  by worth 12,000?

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• 7) Carl plans to invest 500 at 8.25 interest,
  compounded continuously. How long will it take
  for his money to triple?

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• Assignment 10.7
• Page 634 (6, 7, 9, 10, 12) 17, 18
• Lesson 10-7 Page 900 (1-5 all)

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Unit 10 ReviewExploring Exponential and
Logarithmic Functions
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• Unit 10 Test is worth 100 points 8.5 points each
• 12 questions
• Covers sections 10.1 10.7
• Study notes and hw
• Test 10 Review
• Page 638 (1, 3, 4, 6), (9-37 odd), (41-73 odd),
  76, 77, 78

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• Items on the Test
• Exponential Functions
• Logarithmic Form
• Exponential Form
• Properties of Logs
• Product of Logs
• Quotient of Logs
• Power of Logs
• Change of Base

• Common Logarithms
• Antilogarithms
• Natural Logarithms
• Continuously Compound Interest Formula
• Growth and Decay Formula
• Value of Appreciation Depreciation Formula