Precalculus Examples

$8 \log_{2} (\sqrt{3 x - 2}) - \log_{2} (\frac{4}{x})$

Step 1

Simplify each term.

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Step 1.1

Simplify $8 \log_{2} (\sqrt{3 x - 2})$ by moving $8$ inside the logarithm.

$\log_{2} ({\sqrt{3 x - 2}}^{8}) - \log_{2} (\frac{4}{x})$

Step 1.2

Rewrite ${\sqrt{3 x - 2}}^{8}$ as ${(3 x - 2)}^{4}$ .

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Step 1.2.1

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

$\log_{2} ({({(3 x - 2)}^{\frac{1}{2}})}^{8}) - \log_{2} (\frac{4}{x})$

Step 1.2.2

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

$\log_{2} ({(3 x - 2)}^{\frac{1}{2} \cdot 8}) - \log_{2} (\frac{4}{x})$

Step 1.2.3

Combine $\frac{1}{2}$ and $8$ .

$\log_{2} ({(3 x - 2)}^{\frac{8}{2}}) - \log_{2} (\frac{4}{x})$

Step 1.2.4

Cancel the common factor of $8$ and $2$ .

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Step 1.2.4.1

Factor $2$ out of $8$ .

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

Step 1.2.4.2

Cancel the common factors.

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Step 1.2.4.2.1

Factor $2$ out of $2$ .

$\log_{2} ({(3 x - 2)}^{\frac{2 \cdot 4}{2 (1)}}) - \log_{2} (\frac{4}{x})$

Step 1.2.4.2.2

Cancel the common factor.

$\log_{2} ({(3 x - 2)}^{\frac{2 \cdot 4}{2 \cdot 1}}) - \log_{2} (\frac{4}{x})$

Step 1.2.4.2.3

Rewrite the expression.

$\log_{2} ({(3 x - 2)}^{\frac{4}{1}}) - \log_{2} (\frac{4}{x})$

Step 1.2.4.2.4

Divide $4$ by $1$ .

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

Step 2

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

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Combine ${(3 x - 2)}^{4}$ and $\frac{x}{4}$ .

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

Step 5

Reorder factors in $\log_{2} (\frac{{(3 x - 2)}^{4} x}{4})$ .

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