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9.1 EXPONENTIAL FUNCTIONS

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EXPONENTIAL FUNCTIONS A function of the form y=ab x, where a=0, b>0 and b=1. Characteristics 1. continuous and one-to-one 2. domain is the set of all real numbers 3. Range is either all real positive numbers or all real negative numbers depending on whether a is 0 4. x-axis is a horizontal asymptote 5.y-intercept is at a 6. y=ab x and y=a(1/b) x are reflections across the y-axis

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EXAMPLE 1 Sketch the graph of y=2 x. State the domain and range.

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EXAMPLE 2

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EXPONENTIAL GROWTH & DECAY Exponential Growth: Exponential function with base greater than one. y=2(3 x ) Exponential Decay: Exponential function with base between 0 and 1 y=4(1/3) x

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EXAMPLE 3-6 Determine if each function is exponential growth or decay y=(1/5) x y=7(1.2) x y=2(5) x y=10(4/3) x

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STEPS TO WRITE AN EXPONENTIAL FUNCTION 1. Use the y-intercept to find a 2. Choose a second point on the graph to substitute into the equation for x and y. Solve for b. 3. Write your equation in terms of y=ab x (plug in a and b)

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EXAMPLE 7 Write an exponential function using the points (0, 3) and (-1, 6)

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EXAMPLE 8 Write an exponential function using the points (0, -18) and (-2, -2)

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EXAMPLE 9 In 2000, the population of Phoenix was 1,321,045 and it increased to 1,331,391 in A. Write an exponential function of the form y=ab x that could be used to model the population y of Phoenix. Write the function in terms of x, the number of years since B. Suppose the population of Phoenix continues to increase at the same rate. Estimate the population in 2015.

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EXPONENTIAL EQUATIONS Exponential equation: An equation in which the variables are exponents Property of Equality If the base is a number other than 1 and the base is the same, then the two exponents equal each other. 2 x = 2 8 then x=8

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STEPS TO SOLVE EXPONENTIAL EQUATIONS/INEQUALITIES 1. Rewrite the equation so all terms have like bases (you may need to use negative exponents) 2. Set the exponents equal to each other 3. Solve 4. Plug x back in to the original equation to make sure the answer works

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EXAMPLE 10 Solve 3 2n+1 = 81

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EXAMPLE 11 Solve 3 5x = 9 2x-1

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EXAMPLE 12 Solve 4 2x = 8 x-1

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EXAMPLE 13 Solve

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EXAMPLE 14 Solve

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EXAMPLE 15 Solve

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9.2 LOGARITHMS AND LOGARITHMIC FUNCTIONS

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Logarithms with base b Say: “Log base b of x equals y.”

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LOGARITHMIC TO EXPONENTIAL FORM

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EXPONENTIAL TO LOGARITHMIC FORM

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EVALUATE LOGARITHMIC EXPRESSIONS

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CHARACTERISTICS OF LOGARITHMIC FUNCTIONS 1. Inverse of the exponential function y=b x 2.Continous and one-to-one 3. Domain is all positive real numbers and range is ARN 4. y-axis is an asymptote 5. Contains (1,0), so x-intercept is 1

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HELPFUL HINT Since exponential and logarithmic functions are inverses if the bases are the same they “undo” each other…

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LOGARITHMIC EQUATIONS Property of Equality If b is a positive number other than 1, then if and only if x = y.

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EXAMPLE 9 Solve

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EXAMPLE 10 Solve

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EXAMPLE 11 Solve

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LOGARITHMIC TO EXPONENTIAL INEQUALITY If b > 1, x > 0 and log b x > y then x > b y If b > 1, x > 0 and log b x < y then 0< x < b y

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EXAMPLE 12 Solve

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EXAMPLE 13 Solve

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PROPERTY OF INEQUALITY FOR LOGARITHMIC FUNCTIONS If b>1, then if and only if x>y and if and only if x

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EXAMPLE 14

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EXAMPLE 15

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9.3 PROPERTIES OF LOGARITHMS

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PRODUCT PROPERTY The logarithm of a product is the sum of the logarithm of its factors

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QUOTIENT PROPERTY The logarithm of a quotient is the difference of the logarithms of the numerator and denominator.

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POWER PROPERTY The logarithm of a power is the product of the logarithm and the exponent

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EXAMPLE 1

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EXAMPLE 2

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EXAMPLE 3

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EXAMPLE 4

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EXAMPLE 5

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EXAMPLE 6

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9.4 COMMON LOGARITHMS

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COMMON LOGARITHMS Logarithms with base 10 are common logs You do not need to write the 10 it is understood Button on calculator for common logs LOG

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EXAMPLES: USE CALCULATOR TO EVALUATE EACH LOG TO FOUR DECIMAL PLACES 1. log 32. log log 54. log 0.5

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SOLVE LOGARITHMIC EQUATIONS Example 5: The amount of energy E, in ergs, that an earthquake releases is related to is Richter scale magnitude M by the equation logE = M. The Chilean earthquake of 1960 measured 8.5 on the Richter scale. How much energy was released?

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Example 6: Find the energy released by the 2004 Sumatran earthquake, which measured 9.0 on the Richter scale and led to the tsunami.

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HELPFUL HINT If both sides of the equation cannot be easily written as powers of the same base you can solve by taking the log of each side!

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EXAMPLE 3 x =114 x =15

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SOLVING INEQUALITIES Example 7 5 3y <8 y-1

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EXAMPLE 8 3 2x >6 x+1

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EXAMPLE 9 4 y <5 2y+1

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CHANGE OF BASE FORMULA

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EXAMPLE Express in terms of common logs, and then approximate its value to four decimal places. log 4 25log 3 18 log7 5

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9.5 BASE E AND NATURAL LOGS

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NATURAL BASE EXPONENTIAL FUNCTION An exponential function with base e e is the irrational number … *These are used extensively in science to model quantities that grow and decay continuously Calculator button exex

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EVALUATE TO FOUR DECIMAL PLACES 1. e 2 2. e e 1/2

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THE LOG WITH BASE E IS A NATURAL LOG Written as : ln y=ln x is the inverse of y = e x All properties for logs apply the same way to natural logs Calculator button lnx

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EXAMPLES Use calculator to evaluate to four decimal places 4. ln45. ln ln7

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EXAMPLE Write an equivalent exponential or log equation to the given equation. 7. e x =58. lnx≈0.6931

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REMEMBER….. All log properties apply to natural logs Do the same thing for ln problems that you do for log problems Let’s solve!!!!!!!!!

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EXAMPLE 9 Solve e 4x =120 and round to four decimal places

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EXAMPLE 10 EXAMPLE 11 e x

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EXAMPLE 12EXAMPLE 13 ln5x+ln3x>92e 3x + 5 =2

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9.6 EXPONENTIAL GROWTH AND DECAY

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EQUATIONS THAT DEAL WITH E Continuously Compounded Interest A=Pe rt A= amount in account after t years t= # of years r= annual interest rate P= amount of principal invested

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EXAMPLES Suppose you deposit $1000 in an account paying 2.5% annual interest, compounded continuously. Find the balance after 10 years Find how long it will take for the balance to reach at least $1500

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Suppose you deposit $5000 in an account paying 3% annual interst, compounded continuously. Find what the balance would be after 5 years Find how long it will take for the balance to reach at least $7000

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EXPONENTIAL DECAY y=a(1-r) t a=initial amount r=% of decrease expressed as a decimal, this is also called rate of decay t=time y=ae -kt a=initial amount k=constant t=time

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EXAMPLE 3 A cup of coffee contains 130 milligrams of caffeine. If caffeine is eliminated from the body at a rate of 11% per hour, how long will it take for half of this caffeine to be eliminated?

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EXAMPLE 4 The half-life of Sodium-22 is 2.6 years. What is the value of k and the equation of decay for Sodium-22? A geologist examining a meteorite estimates that it contains only about 10% as much Sodium-22 as it would have contained when it reached Earth’s surface. How long ago did the meteorite reach Earth?

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EXPONENTIAL GROWTH y=a(1+r) t a= initial amount r=% of increase/growth expressed as a decimal t=time y=ae kt a=initial amount k=constant t=time

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EXAMPLE 5 Home values in Millersport increase about 4% per year. Mr. Thomas purchased his home eight years ago for $122,000. What is the value of his home now?

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EXAMPLE 6 The population of a city of one million is increasing at a rate of 3% per year. If the population continues to grow at this rate, in how many years will the population have doubled?

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EXAMPLE 7 Two different types of bacteria in two different cultures reproduce exponentially. The first type can be modeled by B 1 (t)=1200e t and the second can be modeled B 2 (t)=3000e t where t is the number of hours. According to these models, how many hours will it take for the amount of B 1 to exceed the amount of B 2 ?

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