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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 |
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| How to Solve a Problem Like: a^x=b |
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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 |
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| Example Problem of a^x=b: 25^x=125 |
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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 |
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| Example Problem of a^(-|x|)=b: 5^(-|x|)=125 |
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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 |
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| How to Solve a Problem Like: a^(-|x|)=b |
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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. |
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| y=log(a)x is equivalent to... |
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Definition
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Example of a Problem Like y=log(a)x:
log(2)x=-2 |
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Definition
1.)Apply y=log(a)x is equivalent to a^y=x
2^(-2)=x
2.)Solve for x
x=.25 |
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How to Solve Exponential Equations Like:
a^(bx+c)=d |
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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 |
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Exponential Equation Example:
2^(7x+3)=23 |
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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 |
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| Terms in an expression may be expanded in a particular way to form an equivalent expression |
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Exponential Equation Example:
17/(4-2^x)=5 |
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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 |
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How to Solve an Exponential Equation Like:
a/(b-c^x)=d |
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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 |
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| Quotient Property of Logarithms |
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Definition
| log(a)M/N=log(a)M-log(a)N |
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Logarithmic Equation Example:
log((x^2)+6x-6)=0 |
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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} |
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How to Solve A Logarithmic Equation Like:
log((x^2)+ax-b)=0 |
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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 |
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Definition
| x=-b(+/-)sqrt((b^2)-4ac)/2a |
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How to Solve Logarithmic Equations Like:
log(a)x+b-log(a)x-c=d |
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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 |
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Logarithmic Equation Example:
log(4)x+54-log(4)x-9=3 |
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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 |
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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. |
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log(a)x=log(a)y
Is Equivalent to..
x=y |
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Definition
| One to One Property of Logarithms |
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Definition
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Example Logarithmic Function:
log(3)2x-7-log(3)4x-1=2 |
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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 |
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Example of Applying a Change of Base Formula:
log(2)x+log(4)x=6 |
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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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