Precalculus Examples

$8 \log_{2} (\sqrt{6 x - 5}) - \log_{2} (\frac{6}{x}) + \log_{2} (6)$

Step 1

Simplify each term.

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Step 1.1

Simplify $8 \log_{2} (\sqrt{6 x - 5})$ by moving $8$ inside the logarithm.

$\log_{2} ({\sqrt{6 x - 5}}^{8}) - \log_{2} (\frac{6}{x}) + \log_{2} (6)$

Step 1.2

Rewrite ${\sqrt{6 x - 5}}^{8}$ as ${(6 x - 5)}^{4}$ .

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 x - 5}$ as ${(6 x - 5)}^{\frac{1}{2}}$ .

$\log_{2} ({({(6 x - 5)}^{\frac{1}{2}})}^{8}) - \log_{2} (\frac{6}{x}) + \log_{2} (6)$

Step 1.2.2

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

$\log_{2} ({(6 x - 5)}^{\frac{1}{2} \cdot 8}) - \log_{2} (\frac{6}{x}) + \log_{2} (6)$

Step 1.2.3

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

$\log_{2} ({(6 x - 5)}^{\frac{8}{2}}) - \log_{2} (\frac{6}{x}) + \log_{2} (6)$

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 $8$ .

$\log_{2} ({(6 x - 5)}^{\frac{2 \cdot 4}{2}}) - \log_{2} (\frac{6}{x}) + \log_{2} (6)$

Step 1.2.4.2

Cancel the common factors.

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Step 1.2.4.2.1

Factor $2$ out of $2$ .

$\log_{2} ({(6 x - 5)}^{\frac{2 \cdot 4}{2 (1)}}) - \log_{2} (\frac{6}{x}) + \log_{2} (6)$

Step 1.2.4.2.2

Cancel the common factor.

$\log_{2} ({(6 x - 5)}^{\frac{2 \cdot 4}{2 \cdot 1}}) - \log_{2} (\frac{6}{x}) + \log_{2} (6)$

Step 1.2.4.2.3

Rewrite the expression.

$\log_{2} ({(6 x - 5)}^{\frac{4}{1}}) - \log_{2} (\frac{6}{x}) + \log_{2} (6)$

Step 1.2.4.2.4

Divide $4$ by $1$ .

$\log_{2} ({(6 x - 5)}^{4}) - \log_{2} (\frac{6}{x}) + \log_{2} (6)$

Step 2

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

$\log_{2} (\frac{{(6 x - 5)}^{4}}{\frac{6}{x}}) + \log_{2} (6)$

Step 3

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

$\log_{2} (\frac{{(6 x - 5)}^{4}}{\frac{6}{x}} \cdot 6)$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\log_{2} ({(6 x - 5)}^{4} \frac{x}{6} \cdot 6)$

Step 5

Combine ${(6 x - 5)}^{4}$ and $\frac{x}{6}$ .

$\log_{2} (\frac{{(6 x - 5)}^{4} x}{6} \cdot 6)$

Step 6

Cancel the common factor of $6$ .

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Step 6.1

Cancel the common factor.

$\log_{2} (\frac{{(6 x - 5)}^{4} x}{6} \cdot 6)$

Step 6.2

Rewrite the expression.

$\log_{2} ({(6 x - 5)}^{4} x)$

Step 7

Reorder factors in $\log_{2} ({(6 x - 5)}^{4} x)$ .

$\log_{2} (x {(6 x - 5)}^{4})$