# Shared Flashcard Set

## Details

Precalculus: 4.4 Homework Notes
Precalculus:4.4 Homework Notes
28
Mathematics
11/05/2015

Term
 One-To-One Property
Definition
 Part One: a^x=a^yPart Two: Is Equivalent to x=yComplete Property: a^x=a^y is equivalent to x=y
Term
 How to Solve a Problem Like: a^x=b
Definition
 1.) Set each side as exponents with the same base (Apply Part One of the One-to-One Property)2.) Make Sure Your Exponents Are Correct on Both Sides of The Equation, the one on the left should have a variable.3.) Create a new equation in which you set the exponents of both sides equal to each other by applying the One-to-One Property.4.) Solve for x
Term
 Example Problem of a^x=b: 25^x=125
Definition
 1.)Apply Part One of One-to-One Property (5^2)^x=5^32.)Make Sure Your Exponents Are Correct on Both Sides of The Equation, the one on the left should have a variable.5^(2x)=5^33.)Apply the One-to-One Property2x=34.)Solve for xx=3/2
Term
 Example Problem of a^(-|x|)=b: 5^(-|x|)=125
Definition
 1.) Apply Part One of One-to-One Property If Able To The Right Side of The Equation5^(-|x|)=5^32.)Apply the One-to-One Property-|x|=33.)Multiply Both Sides By Negative One|x|=-34.)The Solution Is Negative, So it is an Empty Set
Term
 How to Solve a Problem Like: a^(-|x|)=b
Definition
 1.) Check to See If You Can Apply Part One of The One to One Power Property, If so Apply It to the Right Side2.) Create a new equation in which you set the exponents of both sides equal to each other by applying the One-to-One Property.3.) Multiply Both Sides By Negative One to Solve For |x|4.) If |x| is positive then that is your answer, if it is negative however the answer is an empty set.
Term
 y=log(a)x is equivalent to...
Definition
 a^y=x
Term
 Example of a Problem Like y=log(a)x:log(2)x=-2
Definition
 1.)Apply y=log(a)x is equivalent to a^y=x2^(-2)=x2.)Solve for xx=.25
Term
 How to Solve Exponential Equations Like:a^(bx+c)=d
Definition
 1.)Take the Natural Log of Both Sides2.)Apply The Power Rule of Logarithms3.)Apply The Distributive Property On The Left Side 4.)Remove and ignore the x until the end5.)Solve for x
Term
 Exponential Equation Example:2^(7x+3)=23
Definition
 1.)Take the Natural Log of Both Sidesln2^(7x+3)=ln232.)Apply The Power Rule of Logarithms(7x+3)ln2=ln233.)Apply The Distributive Property On The Left Side(7xln2)+(3ln2)=ln234.)Remove and ignore the x until the end(x)(7ln2)+(3ln2)=ln235.)Solve for x(x)(7ln2)+(3ln2)=ln23(x)(7ln2)=ln23-3ln2(x)=ln23-3ln2/7ln2x=ln23-3ln2/7ln2
Term
 lnx is Equivalent To...
Definition
 log(e)x
Term
 Power Rule of Logarithms
Definition
 log(a)M^r=rlog(a)M
Term
 Distributive Property
Definition
 Terms in an expression may be expanded in a particular way to form an equivalent expression
Term
 alnb is equivalent to...
Definition
 a(lnb)
Term
 Exponential Equation Example:17/(4-2^x)=5
Definition
 1.)Multiply Both Sides By The Exponential Equation17=5(4-2^x)2.)Distribute on The Right Side17=20-5*2^x3.)Solve for c^x-3=-5*2^x3/5=2^x4.)Take the Natural Logarithm of Both Sidesln3/5=ln2^x5.)Apply the Power Rule of Logarithmsln3/5=xln26.)Apply the Quotient Power of Logarithmsln3-ln5=xln27.)Isolate the x Using Divisionln3-ln5/ln2=x
Term
 How to Solve an Exponential Equation Like:a/(b-c^x)=d
Definition
 1.)Multiply Both Sides By The Exponential Equation2.)Distribute on The Right Side3.)Solve for c^x4.)Take the Natural Logarithm of Both Sides5.)Apply the Power Rule of Logarithms6.)Apply the Quotient Power of Logarithms7.)Isolate the x Using Division
Term
 Quotient Property of Logarithms
Definition
 log(a)M/N=log(a)M-log(a)N
Term
 Logarithmic Equation Example:log((x^2)+6x-6)=0
Definition
 1.)Write it in exponential form remembering that log(x)=log(10)x and x=y if and only if x=a^y(x^2)+6x-6=10^02.)Simplify the Right Side(x^2)+6x-6=13.)Make the Right Side Equal Zero(x^2)+6x-7=04.)Factor It(x^2)+6x-7=0(x-7)(x+1)x=7 x=-15.) Write It As A Solution Set {7,-1}
Term
 How to Solve A Logarithmic Equation Like:log((x^2)+ax-b)=0
Definition
 1.)Write it in exponential form remembering that log(x)=log(10)x and x=y if and only if x=a^y2.)Simplify the Right Side3.)Make the Right Side Equal Zero4.)Factor It5.)Write It As A Solution Set
Term
 log is equivalent to...
Definition
 log(10)x
Term
 x=y if and only if...
Definition
 x=a^y
Term
Definition
 x=-b(+/-)sqrt((b^2)-4ac)/2a
Term
 How to Solve Logarithmic Equations Like:log(a)x+b-log(a)x-c=d
Definition
 1.)Apply The Quotient Property of Logarithms2.)Write in exponential form by applying y=log(a)x is equivalent to a^y=x
Term
 Logarithmic Equation Example:log(4)x+54-log(4)x-9=3
Definition
 1.)Apply The Quotient Property of Logarithmslog(4)[x+54/x-9]=32.)Write in exponential form by applying y=log(a)x is equivalent to a^y=x[x+54/x-9]=4^33.)Solve for xx=10
Term
 Negative Exponent Rule
Definition
 Negative exponent simply means that the base is on the wrong side of the fraction line, so the base must be flipped to the other side.
Term
 log(a)x=log(a)yIs Equivalent to..x=y
Definition
 One to One Property of Logarithms
Term
 a^(x+y)=(a^x)*(a^y)
Definition
 Exponent Product Rule
Term
 Example Logarithmic Function:log(3)2x-7-log(3)4x-1=2
Definition
 1.)Because both logarithms have the same base,set them as one logarithm by using the quotient property of logarithms:log(3)2x-7/4x-1=22.)Apply y=log(a)x is equivalent to a^y=x:(2x-7/4x)-1=93.)Simplify and Solve for Xx=1/17
Term
 Example of Applying a Change of Base Formula:log(2)x+log(4)x=6
Definition
 1.)log(2)x+log(4)x=6 [Objective: Change the base of log(4)x to 2] 2.)log(2)x+log(2)x/log(2)4 This step shows the base change which can be illustrated as:log(a)x+log(b)x=clog(b)x can be changed to base a as:log(a)x+log(a)x/log(a)b
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