Calculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{2} \log_{2} (x^{4})$ by moving $\frac{1}{2}$ inside the logarithm.

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

Step 1.2

Multiply the exponents in ${(x^{4})}^{\frac{1}{2}}$ .

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Step 1.2.1

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

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

Step 1.2.2

Cancel the common factor of $2$ .

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Step 1.2.2.1

Factor $2$ out of $4$ .

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

Step 1.2.2.2

Cancel the common factor.

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

Step 1.2.2.3

Rewrite the expression.

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

Step 1.3

Simplify $\frac{1}{4} \log_{2} (x^{4})$ by moving $\frac{1}{4}$ inside the logarithm.

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

Step 1.4

Multiply the exponents in ${(x^{4})}^{\frac{1}{4}}$ .

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Step 1.4.1

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

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

Step 1.4.2

Cancel the common factor of $4$ .

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Step 1.4.2.1

Cancel the common factor.

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

Step 1.4.2.2

Rewrite the expression.

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

Step 1.5

Simplify.

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

Step 1.6

Multiply $- \frac{1}{6} \log_{2} (x)$ .

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Step 1.6.1

Reorder $\log_{2} (x)$ and $\frac{1}{6}$ .

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

Step 1.6.2

Simplify $\frac{1}{6} \log_{2} (x)$ by moving $\frac{1}{6}$ inside the logarithm.

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

Step 2

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

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

Step 3

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

$\log_{2} (\frac{x^{2} x}{x^{\frac{1}{6}}})$

Step 4

Move $x^{\frac{1}{6}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 5

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

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Step 5.1

Multiply $x^{2}$ by $x$ .

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.2

Add $2$ and $1$ .

$\log_{2} (x^{3} x^{- \frac{1}{6}})$

Step 6

Multiply $x^{3}$ by $x^{- \frac{1}{6}}$ by adding the exponents.

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Step 6.1

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

$\log_{2} (x^{3 - \frac{1}{6}})$

Step 6.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

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

Step 6.3

Combine $3$ and $\frac{6}{6}$ .

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

Step 6.4

Combine the numerators over the common denominator.

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

Step 6.5

Simplify the numerator.

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Step 6.5.1

Multiply $3$ by $6$ .

$\log_{2} (x^{\frac{18 - 1}{6}})$

Step 6.5.2

Subtract $1$ from $18$ .

$\log_{2} (x^{\frac{17}{6}})$