Precalculus Examples

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

Step 1

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

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

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

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

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.2

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 3.3

Solve the equation for $x$ .

Tap for more steps...

Step 3.3.1

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

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

Step 3.3.2

Multiply $2$ by $1$ .

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

Step 3.3.3

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

Tap for more steps...

Step 3.3.3.1

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

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

Step 3.3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.3.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.3.3.2.1.1

Cancel the common factor.

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

Step 3.3.3.2.1.2

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

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

Step 3.3.4

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 3.3.5

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

Tap for more steps...

Step 3.3.5.1

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

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

Step 3.3.5.2

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 3.3.5.3

Combine and simplify the denominator.

Tap for more steps...

Step 3.3.5.3.1

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

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

Step 3.3.5.3.2

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

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

Step 3.3.5.3.3

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

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

Step 3.3.5.3.4

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 3.3.5.3.5

Add $1$ and $1$ .

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

Step 3.3.5.3.6

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

Tap for more steps...

Step 3.3.5.3.6.1

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 3.3.5.3.6.2

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 3.3.5.3.6.3

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

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

Step 3.3.5.3.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.5.3.6.4.1

Cancel the common factor.

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

Step 3.3.5.3.6.4.2

Rewrite the expression.

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

Step 3.3.5.3.6.5

Evaluate the exponent.

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

Step 3.3.5.4

Simplify the numerator.

Tap for more steps...

Step 3.3.5.4.1

Combine using the product rule for radicals.

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

Step 3.3.5.4.2

Multiply $2$ by $5$ .

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

Step 3.3.6

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

Tap for more steps...

Step 3.3.6.1

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

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

Step 3.3.6.2

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

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

Step 3.3.6.3

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 4

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.63245553 \dots$