Trigonometry Examples

Find the Inverse y=1/(3^x)
Step 1
Interchange the variables.
Step 2
Solve for .
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Step 2.1
Rewrite the equation as .
Step 2.2
Multiply both sides by .
Step 2.3
Simplify the left side.
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Step 2.3.1
Cancel the common factor of .
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Step 2.3.1.1
Cancel the common factor.
Step 2.3.1.2
Rewrite the expression.
Step 2.4
Solve for .
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Step 2.4.1
Rewrite the equation as .
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 2.4.3
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
Step 2.4.4
Expand by moving outside the logarithm.
Step 2.4.5
Divide each term in by and simplify.
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Step 2.4.5.1
Divide each term in by .
Step 2.4.5.2
Simplify the left side.
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Step 2.4.5.2.1
Cancel the common factor of .
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Step 2.4.5.2.1.1
Cancel the common factor.
Step 2.4.5.2.1.2
Divide by .
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 the numerator.
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Step 4.2.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 4.2.3.2
Multiply by .
Step 4.2.4
Expand by moving outside the logarithm.
Step 4.2.5
Cancel the common factor of .
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Step 4.2.5.1
Cancel the common factor.
Step 4.2.5.2
Divide by .
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
Simplify the denominator.
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Step 4.3.3.1
Use the change of base rule .
Step 4.3.3.2
Exponentiation and log are inverse functions.
Step 4.3.4
Multiply the numerator by the reciprocal of the denominator.
Step 4.3.5
Multiply by .
Step 4.4
Since and , then is the inverse of .