Section 12.1: EXPONENTIAL FUNCTIONS When you are done with your homework you should be able to Evaluate eponential functions Graph eponential functions Evaluate functions with base e Use compound interest formulas WARM-UP: Solve. If possible, simplify radicals or rationalize denominators. Epress imaginary solutions in the form a bi. 2 2 2 2 2 6 DEFINITION OF AN EXPONENTIAL FUNCTION The eponential function with base is defined by where is a constant other than ( and ) and is any real number. Created by Shannon Martin Gracey 96
Eample 1: Determine if the given function is an eponential function. a. f 3 b. g 4 1 Eample 2: Evaluate the eponential function at 2, 0, and 2. a. f 2 b. g 1 3 Created by Shannon Martin Gracey 97
Eample 3: Sketch the graph of each eponential function. a. f 3 f 3, f b. g 3 g 3, g How are these two graphs related? Created by Shannon Martin Gracey 98
Eample 4: Sketch the graph of each eponential function. a. f 2 f 2, f b. g 2 1 1 g 2, g How are these two graphs related? Created by Shannon Martin Gracey 99
CHARACTERISTICS OF EXPONENTIAL FUNCTIONS OF THE FORM f b 1. The domain of f b consists of all real numbers:. The range of f b consists of all real numbers:. 2. The graphs of all eponential functions of the form f b pass through the point because ( ). The is. 3. If, f b has a graph that goes to the and is an function. The greater the value of, the steeper the. 4. If, f b has a graph that goes to the and is a function. The smaller the value of, the steeper the. 5. The graph of f b approaches, but does not touch, the. The line is a asymptote. Created by Shannon Martin Gracey 100
n 1 1 n 1 2 5 10 100 1000 10000 100000 1000000 n 1000000000 The irrational number, approimately, is called the base. The function is called the eponential function. FORMULAS FOR COMPOUND INTEREST After years, the balance, in an account with principal and annual interest rate (in decimal form) is given by the following formulas: 1. For compounding interest periods per year: 2. For continuous compounding: Created by Shannon Martin Gracey 101
Eample 5: Find the accumulated value of an investment of $5000 for 10 years at an interest rate of 6.5% if the money is a. compounded semiannually: b. compounded monthly: c. compounded continuously: Created by Shannon Martin Gracey 102
Section 12.2: LOGARITHMIC FUNCTIONS When you are done with your homework you should be able to Change from logarithmic to eponential form Change from eponential to logarithmic form Evaluate logarithms Use basic logarithm properties Graph logarithmic functions Find the domain of a logarithmic function Use common logarithms Use natural logarithms WARM-UP: Graph y 2. y 2, y Created by Shannon Martin Gracey 103
DEFINITION OF THE LOGARITHMIC FUNCTION For and,, is equivalent to. The function is the logarithmic function with base. Eample 1: Write each equation in its equivalent eponential form: a. log4 2 b. y log3 81 Eample 2: Write each equation in its equivalent logarithmic form: y a. e 9 b. b 4 16 Eample 3: Evaluate. a. log5 25 b. log81 9 Created by Shannon Martin Gracey 104
BASIC LOGARITHMIC PROPERTIES INVOLVING 1 1. log b b the power to which I raise to get is 2. log 1 b the power to which I raise to get is INVERSE PROPERTIES OF LOGARITHMS For and, 1. log b b 2. b log b Eample 4: Evaluate. a. log6 6 c. log9 1 b. log 12 12 4 log7 24 d. 7 Eample 5: Sketch the graph of each logarithmic function. f log 3 f log3, f Created by Shannon Martin Gracey 105
CHARACTERISTICS OF LOGARITHMIC FUNCTIONS OF THE FORM f log b 1. The domain of f log b consists of all positive real numbers:. The range of f log b consists of all real numbers:. 2. The graphs of all logarithmic functions of the form f log b pass through the point because. The is. There is no. 3. If, f log b has a graph that goes to the and is an function. 4. If, f log b has a graph that goes to the and is a function. 5. The graph of f log b approaches, but does not touch, the. The line is a asymptote. Eample 6: Find the domain. a. f log2 4 b. f log5 1 Created by Shannon Martin Gracey 106
COMMON LOGARITHMS The logarithmic function with base is called the common logarithmic function. The function is usually epressed as. A calculator with a LOG key can be used to evaluate common logarithms. Eample 7: Evaluate. a. log1000 b. log 0.01 PROPERTIES OF COMMON LOGARITHMS 1. log1 2. log10 3. log10 4. log 10 Eample 8: Evaluate. a. 3 log10 b. log7 10 Created by Shannon Martin Gracey 107
NATURAL LOGARITHMS The logarithmic function with base is called the natural logarithmic function. The function is usually epressed as. A calculator with a LN key can be used to evaluate common logarithms. PROPERTIES OF NATURAL LOGARITHMS 1. ln1 2. ln e 3. ln e 4. e ln Eample 9: Evaluate. 1 a. ln b. e 6 ln300 e Eample 10: Find the domain of f ln 4. 2 Created by Shannon Martin Gracey 108
Section 12.3: PROPERTIES OF LOGARITHMS When you are done with your 12.3 homework you should be able to Use the product rule Use the quotient rule Use the power rule Epand logarithmic epressions Condense logarithmic epressions Use the change-of-base property WARM-UP: Simplify. a. 5 5 b. 3 2 2 THE PRODUCT RULE Let,, and be positive real numbers with. The logarithm of a product is the of the. Eample 1: Epand each logarithmic epression. a. log6 6 b. ln Created by Shannon Martin Gracey 109
THE QUOTIENT RULE Let,, and be positive real numbers with. The logarithm of a quotient is the of the. Eample 2: Epand each logarithmic epression. a. 1 log b. log4 2 THE POWER RULE Let and be positive real numbers with, and let be any real number. The logarithm of a number with an is the of the eponent and the of that number. Eample 3: Epand each logarithmic epression. a. 2 log b. log5 Created by Shannon Martin Gracey 110
PROPERTIES FOR EXPANDING LOGARITHMIC EXPRESSIONS For and : 1. = log M log 2. = log M log b b b b N N 3. = p log b M Eample 4: Epand each logarithmic epression. a. 3 log 3 y log 12y b. 4 5 PROPERTIES FOR CONDENSING LOGARITHMIC EXPRESSIONS For and : 1. = log b MN 2. = log b M N 3. = log M p b Created by Shannon Martin Gracey 111
Eample 5: Write as a single logarithm. a. 1 3ln ln 2 4 b. log4 5 12log4 y THE CHANGE-OF-BASE PROPERTY For any logarithmic bases and, and any positive number, The logarithm of with base is equal to the logarithm of with any new base divided by the logarithm of with that new base. Why would we use this property? Created by Shannon Martin Gracey 112
Eample 6: Use common logarithms to evaluate log5 23. Eample 7: Use natural logarithms to evaluate log5 23. What did you find out??? Created by Shannon Martin Gracey 113
Section 12.4: EXPONENTIAL AND LOGARITHMIC EQUATIONS When you are done with your 12.4 homework you should be able to Use like bases to solve eponential equations Use logarithms to solve eponential equations Use eponential form to solve logarithmic equations Use the one-to-one property of logarithms to solve logarithmic equations Solve applied problems involving eponential and logarithmic equations WARM-UP: Solve. 1 2 5 5 SOLVING EXPONENTIAL EQUATIONS BY EXPRESSING EACH SIDE AS A POWER OF THE SAME BASE If, then. 1. Rewrite the equation in the form. 2. Set. 3. Solve for the variable. Created by Shannon Martin Gracey 114
Eample 1: Solve. a. 2 1 10 100 b. 1 3 4 8 USING LOGARITHMS TO SOLVE EXPONENTIAL EQUATIONS 1. Isolate the epression. 2. Take the logarithm on both sides for base. Take the logarithm on both sides for bases other than 10. 3. Simplify using one of the following properties: 4. Solve for the variable. Eample 2: Solve. a. e 2 6 32 b. 3 2 1 5 c. 10 120 Created by Shannon Martin Gracey 115
USING EXPONENTIAL FORM TO SOLVE LOGARITHMIC EQUATIONS 1. Epress the equation in the form. 2. Use the definition of a logarithm to rewrite the equation in eponential form: 3. Solve for the variable. 4. Check proposed solutions in the equation. Include in the solution set only values for which. Eample 3: Solve. a. log3 log3 2 4 b. log log 21 2 Created by Shannon Martin Gracey 116
USING THE ONE-TO-ONE PROPERTY OF LOGARITHMS TO SOLVE LOGARITHMIC EQUATIONS 1. Epress the equation in the form. This form involves a logarithm whose coefficient is on each side of the equation. 2. Use the one-to-one property to rewrite the equation without logarithms: 3. Solve for the variable. 4. Check proposed solutions in the equation. Include in the solution set only values for which and. Eample 4: Solve. a. 2log6 log6 64 0 b. log 5 1 log 2 3 log 2 Created by Shannon Martin Gracey 117
Section 12.5: EXPONENTIAL GROWTH AND DECAY; MODELING DATA When you are done with your 12.5 homework you should be able to Model eponential growth and decay WARM-UP: Solve. Epress the solution set in terms of logarithms. Then use a calculator to obtain a decimal approimation, correct to two decimal places, for the solution. a. 0.065 1250 6250 e b. 7 4e 10273 One of algebra s many applications is to the behavior of variables. This can be done with eponential and models. With eponential growth or decay, quantities grow or decay ate a rate directly to their size. Created by Shannon Martin Gracey 118
EXPONENTIAL GROWTH AND DECAY MODELS The mathematical model for eponential growth or decay is given by If, the function models the amount, or size, of a entity. is the amount, or size, of the growing entity at time, is the amount at time, and is a constant representing the rate. If, the function models the amount, or size, of a entity. is the amount, or size, of the decaying entity at time, is the amount at time, and is a constant representing the rate. Created by Shannon Martin Gracey 119
Eample 1: In 2000, the population of the Palestinians in the West Bank, Gaza Strip, and East Jerusalem was approimately 3.2 million, and by 2050 it is projected to grow to 12 million. kt a. Use the eponential growth model A A0e, in which t is the number of years after 2000, to find an eponential growth function that models the data. b. In which year will the Palestinian population be 9 million? Created by Shannon Martin Gracey 120
Eample 2: A bird species in danger of etinction has a population that is kt decreasing eponentially ( A A0e ). Five years ago the population was at 1400 and today only 1000 of the birds are alive. Once the population drops below 100, the situation will be irreversible. When will this happen? kt Eample 3: Use the eponential growth model, A A0e, to show that the time it ln 3 takes for a population to triple is given by t k. Created by Shannon Martin Gracey 121
Eample 4: The August 1978 issue of National Geographic described the 1964 find of bones of a newly discovered dinosaur weighing 170 pounds, measuring 9 feet, with a 6 inch claw on one toe of each hind foot. The age of the dinosaur was estimated using potassium-40 dating of rocks surrounding the bones. a. Potassium-40 decays eponentially with a half-life of approimately 1.31 billion years. Use the fact that after 1.31 billion years a given amount of Potassium-40 will have decayed to half the original amount to show that 0.52912t the decay model for Potassium-40 is given by A A e, where t is in billions of years. 0 b. Analysis of the rocks surrounding the dinosaur bones indicated that 94.5% of the original amount of Potassium-40 was still present. Let A 0.945A in the model in part (a) and estimate the age of the bones of 0 the dinosaur. EXPRESSING AN EXPONENTIAL MODEL IN BASE e is equivalent to Eample 5: Rewrite the equation in terms of base e. Epress the answer in terms of a natural logarithm and then round to three decimal places. a. y 1000 7.3 b. y 4.5 0.6 Created by Shannon Martin Gracey 122
Section 13.1: THE CIRCLE When you are done with your 13.1 homework you should be able to Warm-up: Write the standard form of a circle s equation Give the center and radius of a circle whose equation is in standard form Convert the general form of a circle s equation to standard form 1. Solve by completing the square. 2 2 6 2 3 2. Identify the verte of the quadratic function f 2 4 1 Created by Shannon Martin Gracey 123