Algebra Examples

$2 \ln (2 x^{2}) = 1$

Step 1

Simplify $2 \ln (2 x^{2})$ .

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Step 1.1

Simplify $2 \ln (2 x^{2})$ by moving $2$ inside the logarithm.

$\ln ({(2 x^{2})}^{2}) = 1$

Step 1.2

Apply the product rule to $2 x^{2}$ .

$\ln (2^{2} {(x^{2})}^{2}) = 1$

Step 1.3

Raise $2$ to the power of $2$ .

$\ln (4 {(x^{2})}^{2}) = 1$

Step 1.4

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

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Step 1.4.1

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

$\ln (4 x^{2 \cdot 2}) = 1$

Step 1.4.2

Multiply $2$ by $2$ .

$\ln (4 x^{4}) = 1$

Step 2

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

$e^{\ln (4 x^{4})} = e^{1}$

Step 3

Rewrite $\ln (4 x^{4}) = 1$ 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 $b^{y} = x$ .

$e^{1} = 4 x^{4}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $4 x^{4} = e^{1}$ .

$4 x^{4} = e$

Step 4.2

Divide each term in $4 x^{4} = e^{1}$ by $4$ and simplify.

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Step 4.2.1

Divide each term in $4 x^{4} = e^{1}$ by $4$ .

$\frac{4 x^{4}}{4} = \frac{e^{1}}{4}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{4 x^{4}}{4} = \frac{e^{1}}{4}$

Step 4.2.2.1.2

Divide $x^{4}$ by $1$ .

$x^{4} = \frac{e^{1}}{4}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify.

$x^{4} = \frac{e}{4}$

Step 4.3

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

$x = \pm \sqrt[4]{\frac{e}{4}}$

Step 4.4

Simplify $\pm \sqrt[4]{\frac{e}{4}}$ .

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Step 4.4.1

Rewrite $\sqrt[4]{\frac{e}{4}}$ as $\frac{\sqrt[4]{e}}{\sqrt[4]{4}}$ .

$x = \pm \frac{\sqrt[4]{e}}{\sqrt[4]{4}}$

Step 4.4.2

Simplify the denominator.

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Step 4.4.2.1

Rewrite $4$ as $2^{2}$ .

$x = \pm \frac{\sqrt[4]{e}}{\sqrt[4]{2^{2}}}$

Step 4.4.2.2

Rewrite $\sqrt[4]{2^{2}}$ as $\sqrt{\sqrt{2^{2}}}$ .

$x = \pm \frac{\sqrt[4]{e}}{\sqrt{\sqrt{2^{2}}}}$

Step 4.4.2.3

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm \frac{\sqrt[4]{e}}{\sqrt{2}}$

Step 4.4.3

Multiply $\frac{\sqrt[4]{e}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt[4]{e}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 4.4.4

Combine and simplify the denominator.

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Step 4.4.4.1

Multiply $\frac{\sqrt[4]{e}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt[4]{e} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 4.4.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt[4]{e} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 4.4.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt[4]{e} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 4.4.4.4

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

$x = \pm \frac{\sqrt[4]{e} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 4.4.4.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt[4]{e} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 4.4.4.6

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

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Step 4.4.4.6.1

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

$x = \pm \frac{\sqrt[4]{e} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 4.4.4.6.2

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

$x = \pm \frac{\sqrt[4]{e} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 4.4.4.6.3

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

$x = \pm \frac{\sqrt[4]{e} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 4.4.4.6.4

Cancel the common factor of $2$ .

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Step 4.4.4.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt[4]{e} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 4.4.4.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt[4]{e} \sqrt{2}}{2^{1}}$

Step 4.4.4.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt[4]{e} \sqrt{2}}{2}$

Step 4.4.5

Simplify the numerator.

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Step 4.4.5.1

Rewrite the expression using the least common index of $4$ .

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Step 4.4.5.1.1

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

$x = \pm \frac{\sqrt[4]{e} \cdot 2^{\frac{1}{2}}}{2}$

Step 4.4.5.1.2

Rewrite $2^{\frac{1}{2}}$ as $2^{\frac{2}{4}}$ .

$x = \pm \frac{\sqrt[4]{e} \cdot 2^{\frac{2}{4}}}{2}$

Step 4.4.5.1.3

Rewrite $2^{\frac{2}{4}}$ as $\sqrt[4]{2^{2}}$ .

$x = \pm \frac{\sqrt[4]{e} \sqrt[4]{2^{2}}}{2}$

Step 4.4.5.2

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt[4]{e \cdot 2^{2}}}{2}$

Step 4.4.5.3

Raise $2$ to the power of $2$ .

$x = \pm \frac{\sqrt[4]{e \cdot 4}}{2}$

Step 4.4.6

Reorder factors in $\pm \frac{\sqrt[4]{e \cdot 4}}{2}$ .

$x = \pm \frac{\sqrt[4]{4 e}}{2}$

Step 4.5

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

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Step 4.5.1

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

$x = \frac{\sqrt[4]{4 e}}{2}$

Step 4.5.2

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

$x = - \frac{\sqrt[4]{4 e}}{2}$

Step 4.5.3

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

$x = \frac{\sqrt[4]{4 e}}{2}, - \frac{\sqrt[4]{4 e}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt[4]{4 e}}{2}, - \frac{\sqrt[4]{4 e}}{2}$

Decimal Form:

$x = 0.90794307 \dots, - 0.90794307 \dots$