Precalculus Examples

$\log_{3} (\frac{3^{2} y^{4}}{\sqrt{3 x}}) + 3 \log_{3} (5)$

Step 1

Simplify each term.

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Step 1.1

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

$\log_{3} (\frac{9 y^{4}}{\sqrt{3 x}}) + 3 \log_{3} (5)$

Step 1.2

Multiply $\frac{9 y^{4}}{\sqrt{3 x}}$ by $\frac{\sqrt{3 x}}{\sqrt{3 x}}$ .

$\log_{3} (\frac{9 y^{4}}{\sqrt{3 x}} \cdot \frac{\sqrt{3 x}}{\sqrt{3 x}}) + 3 \log_{3} (5)$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

Multiply $\frac{9 y^{4}}{\sqrt{3 x}}$ by $\frac{\sqrt{3 x}}{\sqrt{3 x}}$ .

$\log_{3} (\frac{9 y^{4} \sqrt{3 x}}{\sqrt{3 x} \sqrt{3 x}}) + 3 \log_{3} (5)$

Step 1.3.2

Raise $\sqrt{3 x}$ to the power of $1$ .

$\log_{3} (\frac{9 y^{4} \sqrt{3 x}}{{\sqrt{3 x}}^{1} \sqrt{3 x}}) + 3 \log_{3} (5)$

Step 1.3.3

Raise $\sqrt{3 x}$ to the power of $1$ .

$\log_{3} (\frac{9 y^{4} \sqrt{3 x}}{{\sqrt{3 x}}^{1} {\sqrt{3 x}}^{1}}) + 3 \log_{3} (5)$

Step 1.3.4

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

$\log_{3} (\frac{9 y^{4} \sqrt{3 x}}{{\sqrt{3 x}}^{1 + 1}}) + 3 \log_{3} (5)$

Step 1.3.5

Add $1$ and $1$ .

$\log_{3} (\frac{9 y^{4} \sqrt{3 x}}{{\sqrt{3 x}}^{2}}) + 3 \log_{3} (5)$

Step 1.3.6

Rewrite ${\sqrt{3 x}}^{2}$ as $3 x$ .

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Step 1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x}$ as ${(3 x)}^{\frac{1}{2}}$ .

$\log_{3} (\frac{9 y^{4} \sqrt{3 x}}{{({(3 x)}^{\frac{1}{2}})}^{2}}) + 3 \log_{3} (5)$

Step 1.3.6.2

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

$\log_{3} (\frac{9 y^{4} \sqrt{3 x}}{{(3 x)}^{\frac{1}{2} \cdot 2}}) + 3 \log_{3} (5)$

Step 1.3.6.3

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

$\log_{3} (\frac{9 y^{4} \sqrt{3 x}}{{(3 x)}^{\frac{2}{2}}}) + 3 \log_{3} (5)$

Step 1.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.6.4.1

Cancel the common factor.

$\log_{3} (\frac{9 y^{4} \sqrt{3 x}}{{(3 x)}^{\frac{2}{2}}}) + 3 \log_{3} (5)$

Step 1.3.6.4.2

Rewrite the expression.

$\log_{3} (\frac{9 y^{4} \sqrt{3 x}}{{(3 x)}^{1}}) + 3 \log_{3} (5)$

Step 1.3.6.5

Simplify.

$\log_{3} (\frac{9 y^{4} \sqrt{3 x}}{3 x}) + 3 \log_{3} (5)$

Step 1.4

Cancel the common factor of $9$ and $3$ .

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Step 1.4.1

Factor $3$ out of $9 y^{4} \sqrt{3 x}$ .

$\log_{3} (\frac{3 (3 y^{4} \sqrt{3 x})}{3 x}) + 3 \log_{3} (5)$

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

Factor $3$ out of $3 x$ .

$\log_{3} (\frac{3 (3 y^{4} \sqrt{3 x})}{3 (x)}) + 3 \log_{3} (5)$

Step 1.4.2.2

Cancel the common factor.

$\log_{3} (\frac{3 (3 y^{4} \sqrt{3 x})}{3 x}) + 3 \log_{3} (5)$

Step 1.4.2.3

Rewrite the expression.

$\log_{3} (\frac{3 y^{4} \sqrt{3 x}}{x}) + 3 \log_{3} (5)$

Step 1.5

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

$\log_{3} (\frac{3 y^{4} \sqrt{3 x}}{x}) + \log_{3} (5^{3})$

Step 1.6

Raise $5$ to the power of $3$ .

$\log_{3} (\frac{3 y^{4} \sqrt{3 x}}{x}) + \log_{3} (125)$

Step 2

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

$\log_{3} (\frac{3 y^{4} \sqrt{3 x}}{x} \cdot 125)$

Step 3

Multiply $\frac{3 y^{4} \sqrt{3 x}}{x} \cdot 125$ .

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Step 3.1

Combine $\frac{3 y^{4} \sqrt{3 x}}{x}$ and $125$ .

$\log_{3} (\frac{3 y^{4} \sqrt{3 x} \cdot 125}{x})$

Step 3.2

Multiply $125$ by $3$ .

$\log_{3} (\frac{375 y^{4} \sqrt{3 x}}{x})$