Calculus Examples

$\log_{3} (x^{2}) = 2 \log_{3} (4) - 4 \log_{3} (5)$

Step 1

Simplify the right side.

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Step 1.1

Simplify $2 \log_{3} (4) - 4 \log_{3} (5)$ .

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Simplify $2 \log_{3} (4)$ by moving $2$ inside the logarithm.

$\log_{3} (x^{2}) = \log_{3} (4^{2}) - 4 \log_{3} (5)$

Step 1.1.1.2

Raise $4$ to the power of $2$ .

$\log_{3} (x^{2}) = \log_{3} (16) - 4 \log_{3} (5)$

Step 1.1.1.3

Simplify $- 4 \log_{3} (5)$ by moving $4$ inside the logarithm.

$\log_{3} (x^{2}) = \log_{3} (16) - \log_{3} (5^{4})$

Step 1.1.1.4

Raise $5$ to the power of $4$ .

$\log_{3} (x^{2}) = \log_{3} (16) - \log_{3} (625)$

Step 1.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{3} (x^{2}) = \log_{3} (\frac{16}{625})$

Step 2

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

$x^{2} = \frac{16}{625}$

Step 3

Solve for $x$ .

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Step 3.1

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

$x = \pm \sqrt{\frac{16}{625}}$

Step 3.2

Simplify $\pm \sqrt{\frac{16}{625}}$ .

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Step 3.2.1

Rewrite $\sqrt{\frac{16}{625}}$ as $\frac{\sqrt{16}}{\sqrt{625}}$ .

$x = \pm \frac{\sqrt{16}}{\sqrt{625}}$

Step 3.2.2

Simplify the numerator.

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Step 3.2.2.1

Rewrite $16$ as $4^{2}$ .

$x = \pm \frac{\sqrt{4^{2}}}{\sqrt{625}}$

Step 3.2.2.2

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

$x = \pm \frac{4}{\sqrt{625}}$

Step 3.2.3

Simplify the denominator.

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Step 3.2.3.1

Rewrite $625$ as $25^{2}$ .

$x = \pm \frac{4}{\sqrt{25^{2}}}$

Step 3.2.3.2

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

$x = \pm \frac{4}{25}$

Step 3.3

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

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Step 3.3.1

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

$x = \frac{4}{25}$

Step 3.3.2

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

$x = - \frac{4}{25}$

Step 3.3.3

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

$x = \frac{4}{25}, - \frac{4}{25}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{25}, - \frac{4}{25}$

Decimal Form:

$x = 0.16, - 0.16$