Solve for r natural log of 4r^2=3

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

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 $r$ .

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

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

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

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

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

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

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

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

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

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

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

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

Simplify the numerator.

Factor out $e^{2}$ .

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

Pull terms out from under the radical.

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

Simplify the denominator.

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

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

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

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

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

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

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

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

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

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:

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

Solve for r natural log of 4r^2=3

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