Trigonometry Examples

Find the Inverse y=2 natural log of x-5
Step 1
Interchange the variables.
Step 2
Solve for .
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Step 2.1
Rewrite the equation as .
Step 2.2
Divide each term in by and simplify.
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Step 2.2.1
Divide each term in by .
Step 2.2.2
Simplify the left side.
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Step 2.2.2.1
Cancel the common factor of .
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Step 2.2.2.1.1
Cancel the common factor.
Step 2.2.2.1.2
Divide by .
Step 2.3
To solve for , rewrite the equation using properties of logarithms.
Step 2.4
Rewrite in exponential form using the definition of a logarithm. If and are positive real numbers and , then is equivalent to .
Step 2.5
Solve for .
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Step 2.5.1
Rewrite the equation as .
Step 2.5.2
Add to both sides of the equation.
Step 3
Replace with to show the final answer.
Step 4
Verify if is the inverse of .
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Step 4.1
To verify the inverse, check if and .
Step 4.2
Evaluate .
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Step 4.2.1
Set up the composite result function.
Step 4.2.2
Evaluate by substituting in the value of into .
Step 4.2.3
Simplify each term.
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Step 4.2.3.1
Cancel the common factor of .
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Step 4.2.3.1.1
Cancel the common factor.
Step 4.2.3.1.2
Divide by .
Step 4.2.3.2
Exponentiation and log are inverse functions.
Step 4.2.4
Combine the opposite terms in .
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Step 4.2.4.1
Add and .
Step 4.2.4.2
Add and .
Step 4.3
Evaluate .
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Step 4.3.1
Set up the composite result function.
Step 4.3.2
Evaluate by substituting in the value of into .
Step 4.3.3
Combine the opposite terms in .
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Step 4.3.3.1
Subtract from .
Step 4.3.3.2
Add and .
Step 4.3.4
Use logarithm rules to move out of the exponent.
Step 4.3.5
The natural logarithm of is .
Step 4.3.6
Multiply by .
Step 4.3.7
Cancel the common factor of .
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Step 4.3.7.1
Cancel the common factor.
Step 4.3.7.2
Rewrite the expression.
Step 4.4
Since and , then is the inverse of .