Precalculus Examples

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

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.

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

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

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 $8$ .

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

Step 1.2.4.2

Cancel the common factors.

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Step 1.2.4.2.1

Factor $2$ out of $2$ .

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

Step 1.2.4.2.2

Cancel the common factor.

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

Step 1.2.4.2.3

Rewrite the expression.

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

Step 1.2.4.2.4

Divide $4$ by $1$ .

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

Step 1.3

Logarithm base $2$ of $4$ is $2$ .

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

Step 2

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

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

Step 3

Simplify each term.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

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

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

Step 4

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

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

Step 5

Rewrite $\log_{2} (\frac{x {(3 x - 2)}^{4}}{4})$ using the change of base formula.

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Step 5.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} + 2$

Step 5.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

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

Step 6

To write $2$ as a fraction with a common denominator, multiply by $\frac{\log (2)}{\log (2)}$ .

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Simplify the numerator.

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Step 8.1

Simplify $2 \log (2)$ by moving $2$ inside the logarithm.

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

Cancel the common factor of $4$ .

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Step 8.4.1

Cancel the common factor.

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

Step 8.4.2

Rewrite the expression.

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