Precalculus Examples

$4 \log_{3} (x) + 2 \log_{3} (4 x) - \log_{3} (\sqrt{x})$

Step 1

Simplify each term.

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Step 1.1

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

$\log_{3} (x^{4}) + 2 \log_{3} (4 x) - \log_{3} (\sqrt{x})$

Step 1.2

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

$\log_{3} (x^{4}) + \log_{3} ({(4 x)}^{2}) - \log_{3} (\sqrt{x})$

Step 1.3

Apply the product rule to $4 x$ .

$\log_{3} (x^{4}) + \log_{3} (4^{2} x^{2}) - \log_{3} (\sqrt{x})$

Step 1.4

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

$\log_{3} (x^{4}) + \log_{3} (16 x^{2}) - \log_{3} (\sqrt{x})$

Step 2

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

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

Step 3

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

$\log_{3} (\frac{x^{4} (16 x^{2})}{\sqrt{x}})$

Step 4

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 4.1

Move $x^{2}$ .

$\log_{3} (\frac{x^{2} x^{4} \cdot 16}{\sqrt{x}})$

Step 4.2

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

$\log_{3} (\frac{x^{2 + 4} \cdot 16}{\sqrt{x}})$

Step 4.3

Add $2$ and $4$ .

$\log_{3} (\frac{x^{6} \cdot 16}{\sqrt{x}})$

Step 5

Move $16$ to the left of $x^{6}$ .

$\log_{3} (\frac{16 \cdot x^{6}}{\sqrt{x}})$

Step 6

Multiply $\frac{16 \cdot x^{6}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\log_{3} (\frac{16 \cdot x^{6}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}})$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{16 \cdot x^{6}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\log_{3} (\frac{16 \cdot x^{6} \sqrt{x}}{\sqrt{x} \sqrt{x}})$

Step 7.2

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

$\log_{3} (\frac{16 \cdot x^{6} \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}})$

Step 7.3

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

$\log_{3} (\frac{16 \cdot x^{6} \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}})$

Step 7.4

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

$\log_{3} (\frac{16 \cdot x^{6} \sqrt{x}}{{\sqrt{x}}^{1 + 1}})$

Step 7.5

Add $1$ and $1$ .

$\log_{3} (\frac{16 \cdot x^{6} \sqrt{x}}{{\sqrt{x}}^{2}})$

Step 7.6

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

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Step 7.6.1

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

$\log_{3} (\frac{16 \cdot x^{6} \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}})$

Step 7.6.2

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

$\log_{3} (\frac{16 \cdot x^{6} \sqrt{x}}{x^{\frac{1}{2} \cdot 2}})$

Step 7.6.3

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

$\log_{3} (\frac{16 \cdot x^{6} \sqrt{x}}{x^{\frac{2}{2}}})$

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

$\log_{3} (\frac{16 \cdot x^{6} \sqrt{x}}{x^{\frac{2}{2}}})$

Step 7.6.4.2

Rewrite the expression.

$\log_{3} (\frac{16 \cdot x^{6} \sqrt{x}}{x^{1}})$

Step 7.6.5

Simplify.

$\log_{3} (\frac{16 \cdot x^{6} \sqrt{x}}{x})$

Step 8

Cancel the common factor of $x^{6}$ and $x$ .

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Step 8.1

Factor $x$ out of $16 \cdot x^{6} \sqrt{x}$ .

$\log_{3} (\frac{x (16 \cdot x^{5} \sqrt{x})}{x})$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Raise $x$ to the power of $1$ .

$\log_{3} (\frac{x (16 \cdot x^{5} \sqrt{x})}{x^{1}})$

Step 8.2.2

Factor $x$ out of $x^{1}$ .

$\log_{3} (\frac{x (16 \cdot x^{5} \sqrt{x})}{x \cdot 1})$

Step 8.2.3

Cancel the common factor.

$\log_{3} (\frac{x (16 \cdot x^{5} \sqrt{x})}{x \cdot 1})$

Step 8.2.4

Rewrite the expression.

$\log_{3} (\frac{16 \cdot x^{5} \sqrt{x}}{1})$

Step 8.2.5

Divide $16 \cdot x^{5} \sqrt{x}$ by $1$ .

$\log_{3} (16 \cdot x^{5} \sqrt{x})$

$\log_{3} (16 x^{5} \sqrt{x})$