Calculus Examples

Graph natural log of natural log of x^2
ln(ln(x2))
Step 1
Find the asymptotes.
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Step 1.1
Set the argument of the logarithm equal to zero.
ln(x2)=0
Step 1.2
Solve for x.
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Step 1.2.1
Write in exponential form.
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Step 1.2.1.1
For logarithmic equations, logb(x)=y is equivalent to by=x such that x>0, b>0, and b1. In this case, b=e, x=x2, and y=0.
b=e
x=x2
y=0
Step 1.2.1.2
Substitute the values of b, x, and y into the equation by=x.
e0=x2
e0=x2
Step 1.2.2
Solve for x.
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Step 1.2.2.1
Rewrite the equation as x2=e0.
x2=e0
Step 1.2.2.2
Anything raised to 0 is 1.
x2=1
Step 1.2.2.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
x=±1
Step 1.2.2.4
Any root of 1 is 1.
x=±1
Step 1.2.2.5
The complete solution is the result of both the positive and negative portions of the solution.
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Step 1.2.2.5.1
First, use the positive value of the ± to find the first solution.
x=1
Step 1.2.2.5.2
Next, use the negative value of the ± to find the second solution.
x=-1
Step 1.2.2.5.3
The complete solution is the result of both the positive and negative portions of the solution.
x=1,-1
x=1,-1
x=1,-1
x=1,-1
Step 1.3
The vertical asymptote occurs at x=1,x=-1.
Vertical Asymptote: x=1,x=-1
Vertical Asymptote: x=1,x=-1
Step 2
Find the point at x=-2.
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Step 2.1
Replace the variable x with -2 in the expression.
f(-2)=ln(ln((-2)2))
Step 2.2
Simplify the result.
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Step 2.2.1
Raise -2 to the power of 2.
f(-2)=ln(ln(4))
Step 2.2.2
The final answer is ln(ln(4)).
ln(ln(4))
ln(ln(4))
Step 2.3
Convert ln(ln(4)) to decimal.
=0.32663425
=0.32663425
Step 3
Find the point at x=-3.
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Step 3.1
Replace the variable x with -3 in the expression.
f(-3)=ln(ln((-3)2))
Step 3.2
Simplify the result.
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Step 3.2.1
Raise -3 to the power of 2.
f(-3)=ln(ln(9))
Step 3.2.2
The final answer is ln(ln(9)).
ln(ln(9))
ln(ln(9))
Step 3.3
Convert ln(ln(9)) to decimal.
=0.787195
=0.787195
Step 4
Find the point at x=-4.
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Step 4.1
Replace the variable x with -4 in the expression.
f(-4)=ln(ln((-4)2))
Step 4.2
Simplify the result.
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Step 4.2.1
Raise -4 to the power of 2.
f(-4)=ln(ln(16))
Step 4.2.2
The final answer is ln(ln(16)).
ln(ln(16))
ln(ln(16))
Step 4.3
Convert ln(ln(16)) to decimal.
=1.01978144
=1.01978144
Step 5
The log function can be graphed using the vertical asymptote at x=1,x=-1 and the points (-2,0.32663425),(-3,0.787195),(-4,1.01978144).
Vertical Asymptote: x=1,x=-1
xy-41.02-30.787-20.327
Step 6
 [x2  12  π  xdx ]