Functions - Exponential Functions

1 - Exponential FunctionsObjective: Solve Exponential equations by finding a common our study of algebra gets more advanced we begin to study more involvedfunctions. One pair of inverse Functions we will look at are Exponential functionsand logarithmic Functions . Here we will look at exponentialfunctions and then wewill consider logarithmic Functions in another lesson. Exponential Functions arefunctions where the variable is in the exponent such asf(x) =ax. (It is importantnot to confuse Exponential Functions with polynomial Functions where the variableis in the base such asf(x) =x2).World View NoteOne common application of Exponential Functions is popula-tion growth. According to the 2009 CIA World Factbook, the country with thehighest population growth rate is a tie between the United Arab Emirates (northof Saudi Arabia) and Burundi (central Africa) at There are 32 countrieswith negative growth rates, the lowest being the Northern Mariana Islands (northof Australia) at exponetial equations cannot be done using the skillset we have seen inthe past.

2 For example, if3x= 9, we cannot take thex root of 9 because we donot know what the index is and this doesn t get us any closer tofindingx. How-ever, we may notice that 9 is32. We can then conclude that if3x= 32thenx= is the process we will use to solve Exponential Functions . If we can re-write aproblem so the bases match, then the exponents must also +1=125 Rewrite 125 as5352x+1= 53 Same base,set exponents equal2x+ 1 = 3 Solve 1 1 Subtract1from both sides2x= 2 Divide both sides by222x= 1 Our SolutionSometimes we may have to do work on both sides of the equation to get acommon base. As we do so, we will use various exponent properties to help. Firstwe will use the exponent property that states(ax)y= Rewrite8as23and 32 as25(23)3x= 25 Multiply exponents3and3x29x= 25 Same base,set exponents equal9x= 5 Solve199 Divide both sides by9x=59 Our SolutionAs we multiply exponents we may need to distribute if there are several +5=814x+1 Rewrite 27 as33and 81 as34(92would not be same base)(33)3x+5= (34)4x+1 Multiply exponents3(3x+ 5)and4(4x+ 1)39x+15= 316x+4 Same base,set exponents equal9x+15=16x+ 4 Move variables to one side 9x 9xSubtract9xfrom both sides15= 7x+ 4 Subtract4from both sides 4 411= 7xDivide both sides by777117=xOur SolutionAnother useful exponent property is that negative exponents will give us a recip-rocal,1an=a nExample 4.

3 (19)2x= 37x 1 Rewrite19as3 2(negative exponet to flip)(3 2)2x= 37x 1 Multiply exponents 2and2x3 4x= 37x 1 Same base,set exponets equal 4x= 7x 1 Subtract7xfrom both sides 7x 7x 11x= 1 Divide by 11 11 11x=111 Our SolutionIf we have several factors with the same base on one side of theequation we canadd the exponents using the property that statesaxay=ax+ 52x 1= 53x+11 Add exponents on left,combing like terms56x 1= 53x+11 Same base,set exponents equal6x 1 = 3x+11 Move variables to one sides2 3x 3xSubtract3xfrom both sides3x 1 =11 Add1to both sides+ 1 + 13x=12 Divide both sides by333x= 4 Our SolutionIt may take a bit of practice to get use to knowing which base touse, but as wepractice we will get much quicker at knowing which base to use.

4 As we do so, wewill use our exponent properties to help us simplify. Again,below are the proper-ties we used to simplify.(ax)y=axyand1an=a nandaxay=ax+yWe could see all three properties used in the same problem as we get a commonbase. This is shown in the next 5 (14)3x+1=32 (12)x+3 Write withacommon base of2(24)2x 5 (2 2)3x+1= 25 (2 1)x+3 Multiply exponents,distributing as needed28x 20 2 6x 2= 25 2 x 3 Add exponents,combining like terms22x 22= 2 x+2 Same base,set exponents equal2x 22= x+ 2 Move variables to one side+x+xAddxto both sides3x 22= 2 Add 22 to both sides+22+223x=24 Divide both sides by333x= 8 Our SolutionAll the problems we have solved here we were able to write witha common , not all problems can be written with a common base, for example,2 =10x, we cannot write this problem with a common base.

5 To solve problems likethis we will need to use the inverse of an Exponential function. The inverse iscalled a logarithmic function, which we will discuss in another and Intermediate Algebra by Tyler Wallace is licensed under a Creative CommonsAttribution Unported License. ( ) Practice - Exponential FunctionsSolve each )31 2n= 31 3n3)42a= 15)(125) k=125 2k 27)62m+1=1369)6 3x=3611) 64b= 2513)(14)x=1615)43a= 4317) 363x=2162x+119)92n+3=24321)33x 2= 33x+123)3 2x= 3325)5m+2= 5 m27)(136)b 1=21629)62 2x= 6231)4 2 3n 1=1433)43k 3 42 2k=16 k35)9 2x (1243)3x=243 x37) 64n 2 16n+2= (14)3n 139)5 3n 3 52n= 12)42x=1164) 16 3p=64 3p6) 625 n 2=11258)62r 3= 6r 310)52n= 5 n12) 216 3v=363v14) 27 2n 1= 916)4 3v=6418) 64x+2=1620) 162k=16422) 243p=27 3p24)42n= 42 3n26) 6252x=2528) 2162n=3630)(14)3v 2=641 v32)2166 2a= 63a34) 322p 2 8p= (12)2p36)32m 33m= 138)32 x 33m= 140)43r 4 3r=164 Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative CommonsAttribution Unported License.

6 ( ) - Exponential Functions1)02) 13)04)05) 346) 547) 328)09) 2310)011)5612)013) 214) 5615)116) 117) No solution18) 4319) 1420) 3421) No solution22)023) 3224)2525) 126)1427) 1228)1329)030) No solution31)132)333)1334)2335)036)037)383 8) 139) 340) No solutionBeginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative CommonsAttribution Unported License. ( )5