Calculus Examples

Graph 2 natural log of x
Step 1
Find the asymptotes.
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Step 1.1
Set the argument of the logarithm equal to zero.
Step 1.2
Solve for .
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Step 1.2.1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 1.2.2
Simplify .
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Step 1.2.2.1
Rewrite as .
Step 1.2.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 1.2.2.3
Plus or minus is .
Step 1.3
The vertical asymptote occurs at .
Vertical Asymptote:
Vertical Asymptote:
Step 2
Find the point at .
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Step 2.1
Replace the variable with in the expression.
Step 2.2
Simplify the result.
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Step 2.2.1
The natural logarithm of is .
Step 2.2.2
Multiply by .
Step 2.2.3
The final answer is .
Step 2.3
Convert to decimal.
Step 3
Find the point at .
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Step 3.1
Replace the variable with in the expression.
Step 3.2
Simplify the result.
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Step 3.2.1
Simplify by moving inside the logarithm.
Step 3.2.2
Raise to the power of .
Step 3.2.3
The final answer is .
Step 3.3
Convert to decimal.
Step 4
Find the point at .
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Step 4.1
Replace the variable with in the expression.
Step 4.2
Simplify the result.
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Step 4.2.1
Simplify by moving inside the logarithm.
Step 4.2.2
Raise to the power of .
Step 4.2.3
The final answer is .
Step 4.3
Convert to decimal.
Step 5
The log function can be graphed using the vertical asymptote at and the points .
Vertical Asymptote:
Step 6