Linear Algebra Examples

Find the Domain log base 5 of (4x-5)^2=6
Step 1
Set the argument in greater than to find where the expression is defined.
Step 2
Solve for .
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Step 2.1
Take the specified root of both sides of the inequality to eliminate the exponent on the left side.
Step 2.2
Simplify the equation.
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Step 2.2.1
Simplify the left side.
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Step 2.2.1.1
Pull terms out from under the radical.
Step 2.2.2
Simplify the right side.
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Step 2.2.2.1
Simplify .
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Step 2.2.2.1.1
Rewrite as .
Step 2.2.2.1.2
Pull terms out from under the radical.
Step 2.2.2.1.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 2.3
Write as a piecewise.
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Step 2.3.1
To find the interval for the first piece, find where the inside of the absolute value is non-negative.
Step 2.3.2
Solve the inequality.
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Step 2.3.2.1
Add to both sides of the inequality.
Step 2.3.2.2
Divide each term in by and simplify.
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Step 2.3.2.2.1
Divide each term in by .
Step 2.3.2.2.2
Simplify the left side.
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Step 2.3.2.2.2.1
Cancel the common factor of .
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Step 2.3.2.2.2.1.1
Cancel the common factor.
Step 2.3.2.2.2.1.2
Divide by .
Step 2.3.3
In the piece where is non-negative, remove the absolute value.
Step 2.3.4
To find the interval for the second piece, find where the inside of the absolute value is negative.
Step 2.3.5
Solve the inequality.
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Step 2.3.5.1
Add to both sides of the inequality.
Step 2.3.5.2
Divide each term in by and simplify.
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Step 2.3.5.2.1
Divide each term in by .
Step 2.3.5.2.2
Simplify the left side.
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Step 2.3.5.2.2.1
Cancel the common factor of .
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Step 2.3.5.2.2.1.1
Cancel the common factor.
Step 2.3.5.2.2.1.2
Divide by .
Step 2.3.6
In the piece where is negative, remove the absolute value and multiply by .
Step 2.3.7
Write as a piecewise.
Step 2.3.8
Simplify .
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Step 2.3.8.1
Apply the distributive property.
Step 2.3.8.2
Multiply by .
Step 2.3.8.3
Multiply by .
Step 2.4
Solve for .
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Step 2.4.1
Add to both sides of the inequality.
Step 2.4.2
Divide each term in by and simplify.
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Step 2.4.2.1
Divide each term in by .
Step 2.4.2.2
Simplify the left side.
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Step 2.4.2.2.1
Cancel the common factor of .
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Step 2.4.2.2.1.1
Cancel the common factor.
Step 2.4.2.2.1.2
Divide by .
Step 2.5
Solve for .
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Step 2.5.1
Subtract from both sides of the inequality.
Step 2.5.2
Divide each term in by and simplify.
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Step 2.5.2.1
Divide each term in by . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.
Step 2.5.2.2
Simplify the left side.
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Step 2.5.2.2.1
Cancel the common factor of .
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Step 2.5.2.2.1.1
Cancel the common factor.
Step 2.5.2.2.1.2
Divide by .
Step 2.5.2.3
Simplify the right side.
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Step 2.5.2.3.1
Dividing two negative values results in a positive value.
Step 2.6
Find the union of the solutions.
or
or
Step 3
The domain is all values of that make the expression defined.
Interval Notation:
Set-Builder Notation:
Step 4