Algebra Examples

$\ln (4 r^{2}) = 3$

Step 1

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

$e^{\ln (4 r^{2})} = e^{3}$

Step 2

Rewrite $\ln (4 r^{2}) = 3$ 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 $b^{y} = x$ .

$e^{3} = 4 r^{2}$

Step 3

Solve for $r$ .

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Step 3.1

Rewrite the equation as $4 r^{2} = e^{3}$ .

$4 r^{2} = e^{3}$

Step 3.2

Divide each term in $4 r^{2} = e^{3}$ by $4$ and simplify.

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Step 3.2.1

Divide each term in $4 r^{2} = e^{3}$ by $4$ .

$\frac{4 r^{2}}{4} = \frac{e^{3}}{4}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{4 r^{2}}{4} = \frac{e^{3}}{4}$

Step 3.2.2.1.2

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

$r^{2} = \frac{e^{3}}{4}$

Step 3.3

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

$r = \pm \sqrt{\frac{e^{3}}{4}}$

Step 3.4

Simplify $\pm \sqrt{\frac{e^{3}}{4}}$ .

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Step 3.4.1

Rewrite $\sqrt{\frac{e^{3}}{4}}$ as $\frac{\sqrt{e^{3}}}{\sqrt{4}}$ .

$r = \pm \frac{\sqrt{e^{3}}}{\sqrt{4}}$

Step 3.4.2

Simplify the numerator.

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Step 3.4.2.1

Factor out $e^{2}$ .

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

Step 3.4.2.2

Pull terms out from under the radical.

$r = \pm \frac{e \sqrt{e}}{\sqrt{4}}$

Step 3.4.3

Simplify the denominator.

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Step 3.4.3.1

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

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

Step 3.4.3.2

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

$r = \pm \frac{e \sqrt{e}}{2}$

Step 3.5

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

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Step 3.5.1

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

$r = \frac{e \sqrt{e}}{2}$

Step 3.5.2

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

$r = - \frac{e \sqrt{e}}{2}$

Step 3.5.3

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

$r = \frac{e \sqrt{e}}{2}, - \frac{e \sqrt{e}}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$r = \frac{e \sqrt{e}}{2}, - \frac{e \sqrt{e}}{2}$

Decimal Form:

$r = 2.24084453 \dots, - 2.24084453 \dots$