Ch 8 Alg 2 Note Sheet Key
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1 Ch 8 Alg Note Sheet Key Chapte 8: Eponential and Logaithmic Functions 8. Eploing Eponential Models Fo some data, the est model is a function that uses the independent vaiale as an eponent. An eponential function is a function with the geneal fom y = a, whee is a eal nume, a, >, and. The a is always the initial value (when = ) and it stetches o shinks the paent function. You can use an eponential function with > to model gowth. When >, is the gowth facto. Eample. Gaphing. Eponential Gowth. Gaph y = y 8 8 Apply. Population stats at 8 cats and inceases at a ate of % pe yea. How many cats in yeas? = nume of yeas; y = # of cats ate of incease = % o. gowth facto = % + % = % =. initial value = 8 Model: y = ( initial value)( ) y = 8. ( ) gowth facto y = 8. ( ) 9.5 cats in yeas You can use an eponential function with < < to model decay. When < <, is a decay facto. Eample 5alt. Gaphing. Eponential Decay. Gaph y = y 8 8 Apply. Population stats at 6 cats and deceases at a ate of % pe yea. How many cats in yeas? = nume of yeas; y = # of cats ate of decease = % o. gowth facto = % % = 8% =.8 initial value = 6 Model: y = ( initial value)( ) y = 6(.8) gowth facto y = 6(.8) 6. cats in yeas S. Stiling Page of 5
2 Ch 8 Alg Note Sheet Key Eample. Without gaphing, detemine whethe the function y = (.95) epesents eponential gowth o eponential decay. Decay ecause =.95 and < <. FYI If >, then gowth. Eample 5. Make a tale and gaph y = y = (8) 9 = () 96 = () 8 = () = = 6 = 8 Eample. Refe to the gaph. In, the annual ate of incease in the U.S. population was aout.%. Suppose the ate of incease continues to e.%. Wite a function to models U.S. population gowth, in millions, afte. = nume of yeas afte y = the population in millions gowth facto = % +.% =.% =. model, so fa: y= a( ). Fom gaph when =, y = 8, so 8 a(.) = so 8 a = model: y = ( ) 8. Eample. Wite an eponential function y = a fo a gaph that includes (, ) and (, ). Using (, ): y = a = a a = = a So fa Using (, ): y = a = a a = and since = a= So Find a: = a = and = a a = = = y = S. Stiling Page of 5
3 Ch 8 Alg Note Sheet Key 8. Logaithmic Functions as Inveses Eample. Wite 5 = 5 in aithmic fom. If y =, then y =. If 5 = 5, then 5 5 =. 6 a. 79 =. = 8 c. = Eample. Evaluate = Let the epession =. 6 = 8 Need same ases. = ( ) = = = So 86 = Definition of Logaithm The aithm to the ase of a positive nume y is defined If y =, then y =. y =? ead to what powe is y? Eample a. a. Evaluate = 7 = 9 Need same ases. = ( ) = = = So 9 7 =. Evaluate. = = = =. So A common aithm is a aithm that uses ase. You can wite the common aithm y as y. Note: Thee is a [LOG] key on you calculato. So you can find ase s! Ty numes you know the answes to: (), (), (), (.), (/) Did you get,,, -, -? Why? Ty (-). Did you get ERR:NONREAL ANS? You can t take a of a negative nume! S. Stiling Page of 5
4 Ch 8 Alg Note Sheet Key 8. Popeties of Logaithms Popeties of aithms ae simila to popeties of eponents. Use the connections elow to help you memoize the thee popeties fo aithms. Popety Powes Logaithms (ase o Log) M, N and ae positive and. Poduct s s = + Popety MN = M + N Quotient Popety Powe multiply like ases add eponents Popety ( ) p s = s division like ases sutact eponents = p aise powe to a powe multiply eponents multiplied input add sepaate s (same ase) M = M N divided input sutact sepaate s (same ase) p M = pi M N aise input to a powe multiply y powe Eample. State the popety used. a. 8 =. 8 8 = = Quotient Popety Eample. Epand each aithm. a. 5 y = 5 5 y. y= + y. + y= + y Poduct Popety. = + = + Eample. Simplify each aithm. The goal is to wite it as a single aithm!!. a. = = 5 + y. + y= + y = y. ( ) c Can t do, must have the same ase. S. Stiling Page of 5
5 Ch 8 Alg Note Sheet Key 8.5 Eponential and Logaithmic Equations c An equation of the fom = a, whee the eponent includes a vaiale, is an eponential equation. Equality Popety of Logaithms If m and n ae positive and m = n, then Eample. Solve 7 = 7 = 7 = Take of oth sides. i 7 = Use powe popety. = Divide each side y Use a calculato. Eample. Solve y Gaphing. Solve 6 = 5 Y = 6 and Y = 5 Find the point of intesection on the calculato..8 m= n. Because of the equality popety, you can solve an eponential equation y taking the aithm of each side of the equation. Eample c. + Solve = + = + = Take of oth sides. ( + ) = Use powe popety. + = Divide each side y. = Sutact fom each side..9 Use a calculato. Change of Base Fomula Fo any positive numes M,, and c, with and c, M M = Use this to e ale to ente s of any ase into the calculato! Eample 5 (evised). Find an appoimate value fo =.65 A aithmic equation is an equation that includes a aithmic epession. Eample 6. Solve ( + ) = 5 ( + ) = 5 + = = = 5 + = Wite in eponential fom. Eample 7. Solve = = = Wite as a single. = Wite in eponential fom. = i Don t completely simplify, yet! =± i =± S. Stiling Page 5 of 5
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