Finite Math Examples

Solve for x log base x of y+ log base y of x=2.9 , xy=128
,
Step 1
Solve the equation for .
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Step 1.1
Divide each term in by and simplify.
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Step 1.1.1
Divide each term in by .
Step 1.1.2
Simplify the left side.
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Step 1.1.2.1
Cancel the common factor of .
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Step 1.1.2.1.1
Cancel the common factor.
Step 1.1.2.1.2
Divide by .
Step 1.2
Subtract from both sides of the equation.
Step 1.3
Divide each term in by and simplify.
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Step 1.3.1
Divide each term in by .
Step 1.3.2
Simplify the left side.
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Step 1.3.2.1
Dividing two negative values results in a positive value.
Step 1.3.2.2
Divide by .
Step 1.3.3
Simplify the right side.
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Step 1.3.3.1
Simplify each term.
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Step 1.3.3.1.1
Move the negative one from the denominator of .
Step 1.3.3.1.2
Rewrite as .
Step 1.3.3.1.3
Divide by .
Step 2
Simplify the right side.
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Step 2.1
Simplify .
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Step 2.1.1
Simplify each term.
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Step 2.1.1.1
Simplify the numerator.
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Step 2.1.1.1.1
To write as a fraction with a common denominator, multiply by .
Step 2.1.1.1.2
Combine and .
Step 2.1.1.1.3
Combine the numerators over the common denominator.
Step 2.1.1.1.4
Simplify the numerator.
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Step 2.1.1.1.4.1
Factor out of .
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Step 2.1.1.1.4.1.1
Factor out of .
Step 2.1.1.1.4.1.2
Factor out of .
Step 2.1.1.1.4.1.3
Factor out of .
Step 2.1.1.1.4.2
Simplify by moving inside the logarithm.
Step 2.1.1.1.4.3
Simplify by moving inside the logarithm.
Step 2.1.1.2
Simplify the denominator.
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Step 2.1.1.2.1
To write as a fraction with a common denominator, multiply by .
Step 2.1.1.2.2
Combine and .
Step 2.1.1.2.3
Combine the numerators over the common denominator.
Step 2.1.1.2.4
Simplify the numerator.
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Step 2.1.1.2.4.1
Factor out of .
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Step 2.1.1.2.4.1.1
Factor out of .
Step 2.1.1.2.4.1.2
Factor out of .
Step 2.1.1.2.4.1.3
Factor out of .
Step 2.1.1.2.4.2
Simplify by moving inside the logarithm.
Step 2.1.1.2.4.3
Simplify by moving inside the logarithm.
Step 2.1.1.2.5
To write as a fraction with a common denominator, multiply by .
Step 2.1.1.2.6
Combine and .
Step 2.1.1.2.7
Combine the numerators over the common denominator.
Step 2.1.1.2.8
Simplify the numerator.
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Step 2.1.1.2.8.1
Factor out of .
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Step 2.1.1.2.8.1.1
Factor out of .
Step 2.1.1.2.8.1.2
Factor out of .
Step 2.1.1.2.8.1.3
Factor out of .
Step 2.1.1.2.8.2
Simplify by moving inside the logarithm.
Step 2.1.1.2.8.3
Simplify by moving inside the logarithm.
Step 2.1.2
To write as a fraction with a common denominator, multiply by .
Step 2.1.3
Simplify terms.
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Step 2.1.3.1
Combine and .
Step 2.1.3.2
Combine the numerators over the common denominator.
Step 2.1.4
Simplify the numerator.
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Step 2.1.4.1
Factor out of .
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Step 2.1.4.1.1
Factor out of .
Step 2.1.4.1.2
Factor out of .
Step 2.1.4.1.3
Factor out of .
Step 2.1.4.2
Simplify by moving inside the logarithm.
Step 2.1.4.3
Simplify by moving inside the logarithm.
Step 2.1.5
Simplify with factoring out.
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Step 2.1.5.1
Factor out of .
Step 2.1.5.2
Factor out of .
Step 2.1.5.3
Factor out of .
Step 2.1.5.4
Simplify the expression.
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Step 2.1.5.4.1
Rewrite as .
Step 2.1.5.4.2
Move the negative in front of the fraction.
Step 3
Simplify the right side.
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Step 3.1
Simplify .
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Step 3.1.1
Use the quotient property of logarithms, .
Step 3.1.2
Simplify by moving inside the logarithm.
Step 3.1.3
Simplify the numerator.
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Step 3.1.3.1
Apply the product rule to .
Step 3.1.3.2
Simplify the numerator.
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Step 3.1.3.2.1
Multiply the exponents in .
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Step 3.1.3.2.1.1
Apply the power rule and multiply exponents, .
Step 3.1.3.2.1.2
Multiply by .
Step 3.1.3.2.2
Simplify.
Step 3.1.3.3
Multiply the exponents in .
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Step 3.1.3.3.1
Apply the power rule and multiply exponents, .
Step 3.1.3.3.2
Multiply by .