Notes Algebra 2 Chapter 7 Exponential and Logarithmic Functions. Name. Period. Date 7.1 Graph Exponential Growth Functions

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1 Notes Algera 2 Chapter 7 Eponential and Logarithmic Functions Date 7. Graph Eponential Growth Functions An eponential function has the form = a. Name Period If a > 0 and >, then the function = a is an eponential growth function, and is called the growth factor. An asmptote is a line that a graph approaches more and more closel. When a real-life quantit increases a fied percent each ear (or other time period), the amount of the quantit t ears can e modeled the equation t = a( + r) where a is the initial amount and r is the percent increase epressed as a decimal. Note that +r is the growth factor. Eamples: Graph the function. State the domain and range. ) = 4 2) = 4

2 2 ) = 4) 2 2 = + 5) 4 = 6) 5 =

3 7) =2 + 8) You deposit $000 in an account that pas.2% interest compounded monthl. Find the alance after 2 ears. 9) In 970, the population of Kern Count, CA was aout 0,000. From 970 to 2000, the count population grew at an average annual rate of aout 2.4%. Write an eponential growth model given the population P of Kern Count t ears after 970. Graph the model on our calculator. Compound Interest r A = P + n The amount, A, in the account after t ears with an initial principal, P, deposited in an account that pas interest at an annual rate r (epressed as a decimal), compounded n times per ear. nt Use the graph to estimate the ear when the population of Kern Count was aout 400,000.

4 Date 7.2 Graph Eponential Deca Functions If a > 0 and 0 < <, then the function = a is an eponential deca function, and is called the deca factor. When a real-life quantit decreases a fied percent each ear (or other time Eamples: period), the amount of the quantit t ears can e modeled the equation t = a( r) where a is the initial amount and r is the percent decrease epressed as a decimal. Note that - r is the deca factor. Eamples: Graph the equation. State the domain and range. ) = 2) = 4 4

5 ) 2 = 2 4) = ) = 5 6) 2 = 5

6 + 7) = 4 8) 2 A new all terrain vehicle (ATV) costs $800. The value of the ATV decreases 0% each ear. Write an eponential deca model for the value of the ATV (in dollars) after t ears. Estimate the value after 5 ears. Date 7. Use Functions Involving e The numer denoted the letter e is called the natural ase e or the Euler numer after its discoverer, Leonhard Euler (707-78). The natural ase e is irrational and is defined as follows: As n approaches +, n n + approaches e. Eamples: Simplif the epression. ) 2e 7e e e 2) 2 ) ( 2e ) 4) 5 8 4e 7e 6

7 Graph the function. State the domain and range. 5) = 4e 0.5 2( + ) 6) = e 2 7) 2 + = 5e 2 8) = e 2 7

8 0.22t 9) A population of acteria can e modeled the function P= 70e where t is the time (in hours). Graph the model and use the graph to estimate the population after 4 hours. (Hint: Use our calculator and the trace feature.) Continuousl Compounded Interest When interest is compounded continuousl, the amount A in an account after t ears is given the formula rt A= Pe Where P is the principal and r is the annual interest rate epressed as a decimal. 0) You deposit $500 in an account that pas % annual interest compounded continuousl. What is the alance after 2 ears. Date 7.4 Evaluate Logarithms and Graph Logarithmic Functions Let and e positive numers with. The logarithm of with ase is denoted log and is defined as follows log = if and onl if =. A common logarithm is a logarithm with ase 0, denoted log. A natural logarithm is a logarithm is logarithm with ase, denoted ln. 8

9 Eamples: Rewrite the equation in eponential form. Logarithmic Form Eponential Form log 2 6= log 4 = 0 log 9 9= 4 log / 5 25= 2 Evaluate the logarithm.. 8 log 2. log log / log 27 Use the inverse properties to simplif. log log / 6 6 ln e 8. log log ln e 2 Find the inverse of the function.. = 8 2. = ln( 2). = 5 4. = ln( +) 9

10 Graph 5. = log ( + ) 2 (Hint: First graph the parent function = log and then translate.) Date 7.5 Appl Properties of Logarithms Laws of Logarithms Product Propert: log mn= log m+ log m Quotient Propert: log n n Power Propert: log m = Change of Base Formula loga log a= log = log nlog m log m n n Eamples: Use the properties of logarithms and the fact log and log 6, 6. 6 log 2. log 0 5. log 6 0

11 Epand the logarithmic epression log 8 5. ln z Condense the epression. 6. ln + ln 2 ln log + ½ log 8. Average student scores on a memor eam are modeled the function f( t) = 00 2 log( t+ ) where t is the time in months. a) Use the properties of logarithms to write the model in condensed form. ) Find the average score after months. 7.6 Solve Eponential and Logarithmic Equations Eponential equations are equations in which variale epressions occur as eponents. Logarithmic equations are equations that involve logarithms of variale epressions. Eamples: Solve the equation = = 2. = = 9

12 / 2 5. e 2= = 2 2 Solve the logarithmic equation. 7. log8( + 6) = log8(4 ) 8. log7( + 2) = log7(2 ) ln = ln = ln(2 ) + ln( 2) 2