Calculus Examples

$\ln (\sqrt{x y})$

Step 1

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (\sqrt{x y})} = e^{0}$

Step 2

Rewrite $\ln (\sqrt{x y}) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{0} = \sqrt{x y}$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $\sqrt{x y} = e^{0}$ .

$\sqrt{x y} = e^{0}$

Step 3.2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x y}}^{2} = {(e^{0})}^{2}$

Step 3.3

Simplify each side of the equation.

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Step 3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x y}$ as ${(x y)}^{\frac{1}{2}}$ .

${({(x y)}^{\frac{1}{2}})}^{2} = {(e^{0})}^{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify ${({(x y)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1

Multiply the exponents in ${({(x y)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x y)}^{\frac{1}{2} \cdot 2} = {(e^{0})}^{2}$

Step 3.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.2.1.1.2.1

Cancel the common factor.

${(x y)}^{\frac{1}{2} \cdot 2} = {(e^{0})}^{2}$

Step 3.3.2.1.1.2.2

Rewrite the expression.

${(x y)}^{1} = {(e^{0})}^{2}$

Step 3.3.2.1.2

Simplify.

$x y = {(e^{0})}^{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify ${(e^{0})}^{2}$ .

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Step 3.3.3.1.1

Multiply the exponents in ${(e^{0})}^{2}$ .

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Step 3.3.3.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x y = e^{0 \cdot 2}$

Step 3.3.3.1.1.2

Multiply $0$ by $2$ .

$x y = e^{0}$

Step 3.3.3.1.2

Anything raised to $0$ is $1$ .

$x y = 1$

Step 3.4

Divide each term in $x y = 1$ by $y$ and simplify.

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Step 3.4.1

Divide each term in $x y = 1$ by $y$ .

$\frac{x y}{y} = \frac{1}{y}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $y$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{x y}{y} = \frac{1}{y}$

Step 3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{y}$

Step 4

Find the $y$ value at $x = - 1$ .

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Step 4.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = \frac{1}{- 1}$

Step 4.2

Simplify the result.

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Step 4.2.1

Divide $1$ by $- 1$ .

$f (- 1) = - 1$

Step 4.2.2

The final answer is $- 1$ .

$- 1$

Step 4.3

The $y$ value at $x = - 1$ is $- 1$ .

$y = - 1$

Step 5

Find the $y$ value at $x = - 2$ .

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Step 5.1

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = \frac{1}{- 2}$

Step 5.2

Simplify the result.

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Step 5.2.1

Move the negative in front of the fraction.

$f (- 2) = - \frac{1}{2}$

Step 5.2.2

The final answer is $- \frac{1}{2}$ .

$- \frac{1}{2}$

Step 5.3

The $y$ value at $x = - 2$ is $- \frac{1}{2}$ .

$y = - \frac{1}{2}$

Step 6

Find the $y$ value at $x = - 3$ .

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Step 6.1

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = \frac{1}{- 3}$

Step 6.2

Simplify the result.

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Step 6.2.1

Move the negative in front of the fraction.

$f (- 3) = - \frac{1}{3}$

Step 6.2.2

The final answer is $- \frac{1}{3}$ .

$- \frac{1}{3}$

Step 6.3

The $y$ value at $x = - 3$ is $- \frac{1}{3}$ .

$y = - \frac{1}{3}$

Step 7

Find the $y$ value at $x = 1$ .

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Step 7.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \frac{1}{1}$

Step 7.2

Simplify the result.

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Step 7.2.1

Divide $1$ by $1$ .

$f (1) = 1$

Step 7.2.2

The final answer is $1$ .

$1$

Step 7.3

The $y$ value at $x = 1$ is $1$ .

$y = 1$

Step 8

Find the $y$ value at $x = 2$ .

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Step 8.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = \frac{1}{2}$

Step 8.2

The final answer is $\frac{1}{2}$ .

$\frac{1}{2}$

Step 8.3

The $y$ value at $x = 2$ is $\frac{1}{2}$ .

$y = \frac{1}{2}$

Step 9

Find the $y$ value at $x = 3$ .

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Step 9.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = \frac{1}{3}$

Step 9.2

The final answer is $\frac{1}{3}$ .

$\frac{1}{3}$

Step 9.3

The $y$ value at $x = 3$ is $\frac{1}{3}$ .

$y = \frac{1}{3}$

Step 10

List the points to graph.

$(- 1, - 1), (- 2, - 0.5), (- 3, - 0.\overline{3}), (1, 1), (2, 0.5), (3, 0.\overline{3})$

Step 11

Select a few points to graph.

$\begin{array}{c} x & y \\ - 3 & - 0.333 \\ - 2 & - 0.5 \\ - 1 & - 1 \\ 1 & 1 \\ 2 & 0.5 \\ 3 & 0.333 \end{array}$

Step 12