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Calculus Examples
Step 1
Write as an equation.
Step 2
Interchange the variables.
Step 3
Step 3.1
Rewrite the equation as .
Step 3.2
Multiply both sides by .
Step 3.3
Simplify.
Step 3.3.1
Simplify the left side.
Step 3.3.1.1
Cancel the common factor of .
Step 3.3.1.1.1
Cancel the common factor.
Step 3.3.1.1.2
Rewrite the expression.
Step 3.3.2
Simplify the right side.
Step 3.3.2.1
Simplify .
Step 3.3.2.1.1
Apply the distributive property.
Step 3.3.2.1.2
Simplify the expression.
Step 3.3.2.1.2.1
Multiply by .
Step 3.3.2.1.2.2
Rewrite using the commutative property of multiplication.
Step 3.4
Solve for .
Step 3.4.1
Subtract from both sides of the equation.
Step 3.4.2
Factor out of .
Step 3.4.2.1
Multiply by .
Step 3.4.2.2
Factor out of .
Step 3.4.2.3
Factor out of .
Step 3.4.3
Divide each term in by and simplify.
Step 3.4.3.1
Divide each term in by .
Step 3.4.3.2
Simplify the left side.
Step 3.4.3.2.1
Cancel the common factor of .
Step 3.4.3.2.1.1
Cancel the common factor.
Step 3.4.3.2.1.2
Divide by .
Step 3.4.4
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
Step 3.4.5
Expand the left side.
Step 3.4.5.1
Expand by moving outside the logarithm.
Step 3.4.5.2
The natural logarithm of is .
Step 3.4.5.3
Multiply by .
Step 4
Replace with to show the final answer.
Step 5
Step 5.1
To verify the inverse, check if and .
Step 5.2
Evaluate .
Step 5.2.1
Set up the composite result function.
Step 5.2.2
Evaluate by substituting in the value of into .
Step 5.2.3
Multiply the numerator by the reciprocal of the denominator.
Step 5.2.4
Simplify the denominator.
Step 5.2.4.1
Combine and .
Step 5.2.4.2
Move the negative in front of the fraction.
Step 5.2.4.3
Write as a fraction with a common denominator.
Step 5.2.4.4
Combine the numerators over the common denominator.
Step 5.2.4.5
Reorder terms.
Step 5.2.4.6
Rewrite in a factored form.
Step 5.2.4.6.1
Subtract from .
Step 5.2.4.6.2
Add and .
Step 5.2.5
Multiply the numerator by the reciprocal of the denominator.
Step 5.2.6
Multiply by .
Step 5.2.7
Multiply by .
Step 5.2.8
Cancel the common factor of and .
Step 5.2.8.1
Reorder terms.
Step 5.2.8.2
Cancel the common factor.
Step 5.2.8.3
Divide by .
Step 5.2.9
Use logarithm rules to move out of the exponent.
Step 5.2.10
The natural logarithm of is .
Step 5.2.11
Multiply by .
Step 5.3
Evaluate .
Step 5.3.1
Set up the composite result function.
Step 5.3.2
Evaluate by substituting in the value of into .
Step 5.3.3
Exponentiation and log are inverse functions.
Step 5.3.4
Simplify the denominator.
Step 5.3.4.1
Exponentiation and log are inverse functions.
Step 5.3.4.2
Combine and .
Step 5.3.4.3
Write as a fraction with a common denominator.
Step 5.3.4.4
Combine the numerators over the common denominator.
Step 5.3.4.5
Rewrite in a factored form.
Step 5.3.4.5.1
Add and .
Step 5.3.4.5.2
Add and .
Step 5.3.5
Multiply the numerator by the reciprocal of the denominator.
Step 5.3.6
Cancel the common factor of .
Step 5.3.6.1
Cancel the common factor.
Step 5.3.6.2
Rewrite the expression.
Step 5.4
Since and , then is the inverse of .