Calculus Examples

Solve for x natural log of x+1=2+ natural log of x
Step 1
Move all the terms containing a logarithm to the left side of the equation.
Step 2
Use the quotient property of logarithms, .
Step 3
Rewrite in exponential form using the definition of a logarithm. If and are positive real numbers and , then is equivalent to .
Step 4
Cross multiply to remove the fraction.
Step 5
Multiply by .
Step 6
Subtract from both sides of the equation.
Step 7
Subtract from both sides of the equation.
Step 8
Factor the left side of the equation.
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Step 8.1
Factor out of .
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Step 8.1.1
Raise to the power of .
Step 8.1.2
Factor out of .
Step 8.1.3
Factor out of .
Step 8.1.4
Factor out of .
Step 8.2
Rewrite as .
Step 8.3
Factor.
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Step 8.3.1
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 8.3.2
Remove unnecessary parentheses.
Step 9
Divide each term in by and simplify.
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Step 9.1
Divide each term in by .
Step 9.2
Simplify the left side.
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Step 9.2.1
Simplify the denominator.
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Step 9.2.1.1
Rewrite as .
Step 9.2.1.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 9.2.2
Reduce the expression by cancelling the common factors.
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Step 9.2.2.1
Cancel the common factor of .
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Step 9.2.2.1.1
Cancel the common factor.
Step 9.2.2.1.2
Rewrite the expression.
Step 9.2.2.2
Cancel the common factor of .
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Step 9.2.2.2.1
Cancel the common factor.
Step 9.2.2.2.2
Divide by .
Step 9.3
Simplify the right side.
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Step 9.3.1
Simplify the denominator.
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Step 9.3.1.1
Rewrite as .
Step 9.3.1.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 9.3.2
Move the negative in front of the fraction.
Step 10
The result can be shown in multiple forms.
Exact Form:
Decimal Form: