Algebra Examples

$2 \ln (x) + 3 \ln (2) = 5$

Step 1

Move all the terms containing a logarithm to the left side of the equation.

$2 \ln (x) + 3 \ln (2) = 5$

Step 2

Simplify the left side.

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Step 2.1

Simplify $2 \ln (x) + 3 \ln (2)$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

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

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

Step 2.1.1.2

Simplify $3 \ln (2)$ by moving $3$ inside the logarithm.

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

Step 2.1.1.3

Raise $2$ to the power of $3$ .

$\ln (x^{2}) + \ln (8) = 5$

Step 2.1.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln (x^{2} \cdot 8) = 5$

Step 2.1.3

Move $8$ to the left of $x^{2}$ .

$\ln (8 x^{2}) = 5$

Step 3

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

$e^{\ln (8 x^{2})} = e^{5}$

Step 4

Rewrite $\ln (8 x^{2}) = 5$ 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 $b^{y} = x$ .

$e^{5} = 8 x^{2}$

Step 5

Solve for $x$ .

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Step 5.1

Rewrite the equation as $8 x^{2} = e^{5}$ .

$8 x^{2} = e^{5}$

Step 5.2

Divide each term in $8 x^{2} = e^{5}$ by $8$ and simplify.

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Step 5.2.1

Divide each term in $8 x^{2} = e^{5}$ by $8$ .

$\frac{8 x^{2}}{8} = \frac{e^{5}}{8}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $8$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{8 x^{2}}{8} = \frac{e^{5}}{8}$

Step 5.2.2.1.2

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

$x^{2} = \frac{e^{5}}{8}$

Step 5.3

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

$x = \pm \sqrt{\frac{e^{5}}{8}}$

Step 5.4

Simplify $\pm \sqrt{\frac{e^{5}}{8}}$ .

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Step 5.4.1

Rewrite $\sqrt{\frac{e^{5}}{8}}$ as $\frac{\sqrt{e^{5}}}{\sqrt{8}}$ .

$x = \pm \frac{\sqrt{e^{5}}}{\sqrt{8}}$

Step 5.4.2

Simplify the numerator.

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Step 5.4.2.1

Rewrite $e^{5}$ as ${(e^{2})}^{2} e$ .

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Step 5.4.2.1.1

Factor out $e^{4}$ .

$x = \pm \frac{\sqrt{e^{4} e}}{\sqrt{8}}$

Step 5.4.2.1.2

Rewrite $e^{4}$ as ${(e^{2})}^{2}$ .

$x = \pm \frac{\sqrt{{(e^{2})}^{2} e}}{\sqrt{8}}$

Step 5.4.2.2

Pull terms out from under the radical.

$x = \pm \frac{e^{2} \sqrt{e}}{\sqrt{8}}$

Step 5.4.3

Simplify the denominator.

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Step 5.4.3.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 5.4.3.1.1

Factor $4$ out of $8$ .

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

Step 5.4.3.1.2

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

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

Step 5.4.3.2

Pull terms out from under the radical.

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

Step 5.4.4

Multiply $\frac{e^{2} \sqrt{e}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 5.4.5

Combine and simplify the denominator.

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Step 5.4.5.1

Multiply $\frac{e^{2} \sqrt{e}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 5.4.5.2

Move $\sqrt{2}$ .

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

Step 5.4.5.3

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

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

Step 5.4.5.4

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

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

Step 5.4.5.5

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

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

Step 5.4.5.6

Add $1$ and $1$ .

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

Step 5.4.5.7

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

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Step 5.4.5.7.1

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

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

Step 5.4.5.7.2

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

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

Step 5.4.5.7.3

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

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

Step 5.4.5.7.4

Cancel the common factor of $2$ .

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Step 5.4.5.7.4.1

Cancel the common factor.

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

Step 5.4.5.7.4.2

Rewrite the expression.

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

Step 5.4.5.7.5

Evaluate the exponent.

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

Step 5.4.6

Combine using the product rule for radicals.

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

Step 5.4.7

Multiply $2$ by $2$ .

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

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

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

Step 5.5.3

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

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

Step 6

Exclude the solutions that do not make $2 \ln (x) + 3 \ln (2) = 5$ true.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 4.30716204 \dots$