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Precalculus: 4.4 Homework Notes
Precalculus:4.4 Homework Notes
28
Mathematics
Undergraduate 1
11/05/2015

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Term
One-To-One Property
Definition
Part One: a^x=a^y
Part Two: Is Equivalent to x=y

Complete 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=5^3
2.)Make Sure Your Exponents Are Correct on Both Sides of The Equation, the one on the left should have a variable.
5^(2x)=5^3
3.)Apply the One-to-One Property
2x=3
4.)Solve for x
x=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 Equation
5^(-|x|)=5^3
2.)Apply the One-to-One Property
-|x|=3
3.)Multiply Both Sides By Negative One
|x|=-3
4.)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 Side
2.) 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=x
2^(-2)=x
2.)Solve for x
x=.25
Term
How to Solve Exponential Equations Like:
a^(bx+c)=d
Definition
1.)Take the Natural Log of Both Sides
2.)Apply The Power Rule of Logarithms
3.)Apply The Distributive Property On The Left Side
4.)Remove and ignore the x until the end
5.)Solve for x
Term
Exponential Equation Example:
2^(7x+3)=23
Definition
1.)Take the Natural Log of Both Sides
ln2^(7x+3)=ln23
2.)Apply The Power Rule of Logarithms
(7x+3)ln2=ln23
3.)Apply The Distributive Property On The Left Side
(7xln2)+(3ln2)=ln23
4.)Remove and ignore the x until the end
(x)(7ln2)+(3ln2)=ln23
5.)Solve for x
(x)(7ln2)+(3ln2)=ln23
(x)(7ln2)=ln23-3ln2
(x)=ln23-3ln2/7ln2
x=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 Equation
17=5(4-2^x)
2.)Distribute on The Right Side
17=20-5*2^x
3.)Solve for c^x
-3=-5*2^x
3/5=2^x
4.)Take the Natural Logarithm of Both Sides
ln3/5=ln2^x
5.)Apply the Power Rule of Logarithms
ln3/5=xln2
6.)Apply the Quotient Power of Logarithms
ln3-ln5=xln2
7.)Isolate the x Using Division
ln3-ln5/ln2=x
Term
How to Solve an Exponential Equation Like:
a/(b-c^x)=d
Definition
1.)Multiply Both Sides By The Exponential Equation
2.)Distribute on The Right Side
3.)Solve for c^x
4.)Take the Natural Logarithm of Both Sides
5.)Apply the Power Rule of Logarithms
6.)Apply the Quotient Power of Logarithms
7.)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=10^0
2.)Simplify the Right Side
(x^2)+6x-6=1
3.)Make the Right Side Equal Zero
(x^2)+6x-7=0
4.)Factor It
(x^2)+6x-7=0
(x-7)(x+1)
x=7 x=-1
5.) 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^y
2.)Simplify the Right Side
3.)Make the Right Side Equal Zero
4.)Factor It
5.)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
Quadratic Formula
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 Logarithms
2.)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 Logarithms
log(4)[x+54/x-9]=3
2.)Write in exponential form by applying y=log(a)x is equivalent to a^y=x
[x+54/x-9]=4^3
3.)Solve for x
x=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)y
Is 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=2
2.)Apply y=log(a)x is equivalent to a^y=x:
(2x-7/4x)-1=9
3.)Simplify and Solve for X
x=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=c
log(b)x can be changed to base a as:
log(a)x+log(a)x/log(a)b
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