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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 2004. 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 2000. 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 0.2 3. 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 = 11.8 + 1.5M. 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 2.71828… *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 -1.3 3. 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. ln0.056. 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-2 + 4 4

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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 0.1532t and the second can be modeled B 2 (t)=3000e 0.0466t 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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