Calculus Examples

$e^{\ln (x) - 3 \ln (x)} = \frac{5}{2}$

Step 1

Solve for $x$ .

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Step 1.1

Simplify $e^{\ln (x) - 3 \ln (x)}$ .

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Step 1.1.1

Simplify $- 3 \ln (x)$ by moving $3$ inside the logarithm.

$e^{\ln (x) - \ln (x^{3})} = \frac{5}{2}$

Step 1.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$e^{\ln (\frac{x}{x^{3}})} = \frac{5}{2}$

Step 1.1.3

Exponentiation and log are inverse functions.

$\frac{x}{x^{3}} = \frac{5}{2}$

Step 1.1.4

Cancel the common factor of $x$ and $x^{3}$ .

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Step 1.1.4.1

Raise $x$ to the power of $1$ .

$\frac{x^{1}}{x^{3}} = \frac{5}{2}$

Step 1.1.4.2

Factor $x$ out of $x^{1}$ .

$\frac{x \cdot 1}{x^{3}} = \frac{5}{2}$

Step 1.1.4.3

Cancel the common factors.

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Step 1.1.4.3.1

Factor $x$ out of $x^{3}$ .

$\frac{x \cdot 1}{x \cdot x^{2}} = \frac{5}{2}$

Step 1.1.4.3.2

Cancel the common factor.

$\frac{x \cdot 1}{x \cdot x^{2}} = \frac{5}{2}$

Step 1.1.4.3.3

Rewrite the expression.

$\frac{1}{x^{2}} = \frac{5}{2}$

Step 1.2

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$1 \cdot 2 = x^{2} \cdot 5$

Step 1.3

Solve the equation for $x$ .

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Step 1.3.1

Rewrite the equation as $x^{2} \cdot 5 = 1 \cdot 2$ .

$x^{2} \cdot 5 = 1 \cdot 2$

Step 1.3.2

Multiply $2$ by $1$ .

$x^{2} \cdot 5 = 2$

Step 1.3.3

Divide each term in $x^{2} \cdot 5 = 2$ by $5$ and simplify.

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Step 1.3.3.1

Divide each term in $x^{2} \cdot 5 = 2$ by $5$ .

$\frac{x^{2} \cdot 5}{5} = \frac{2}{5}$

Step 1.3.3.2

Simplify the left side.

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Step 1.3.3.2.1

Cancel the common factor of $5$ .

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Step 1.3.3.2.1.1

Cancel the common factor.

$\frac{x^{2} \cdot 5}{5} = \frac{2}{5}$

Step 1.3.3.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{2}{5}$

Step 1.3.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{2}{5}}$

Step 1.3.5

Simplify $\pm \sqrt{\frac{2}{5}}$ .

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Step 1.3.5.1

Rewrite $\sqrt{\frac{2}{5}}$ as $\frac{\sqrt{2}}{\sqrt{5}}$ .

$x = \pm \frac{\sqrt{2}}{\sqrt{5}}$

Step 1.3.5.2

Multiply $\frac{\sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{\sqrt{2}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 1.3.5.3

Combine and simplify the denominator.

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Step 1.3.5.3.1

Multiply $\frac{\sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{\sqrt{2} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 1.3.5.3.2

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 1.3.5.3.3

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 1.3.5.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 1.3.5.3.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 1.3.5.3.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 1.3.5.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{2} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 1.3.5.3.6.2

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

$x = \pm \frac{\sqrt{2} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 1.3.5.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 1.3.5.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.5.3.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 1.3.5.3.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{2} \sqrt{5}}{5^{1}}$

Step 1.3.5.3.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{2} \sqrt{5}}{5}$

Step 1.3.5.4

Simplify the numerator.

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Step 1.3.5.4.1

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{2 \cdot 5}}{5}$

Step 1.3.5.4.2

Multiply $2$ by $5$ .

$x = \pm \frac{\sqrt{10}}{5}$

Step 1.3.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.3.6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{10}}{5}$

Step 1.3.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{10}}{5}$

Step 1.3.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{10}}{5}, - \frac{\sqrt{10}}{5}$

Step 2

Exclude the solutions that do not make $e^{\ln (x) - 3 \ln (x)} = \frac{5}{2}$ true.

$x = \frac{\sqrt{10}}{5}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{10}}{5}$

Decimal Form:

$x = 0.63245553 \dots$