Algebra Examples

$\frac{1}{2} \cdot \log_{2} (9 x^{16}) - \frac{2}{3} \cdot \log_{2} (8 x^{3})$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{2} \log_{2} (9 x^{16})$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{2} ({(9 x^{16})}^{\frac{1}{2}}) - \frac{2}{3} \cdot \log_{2} (8 x^{3})$

Step 1.2

Apply the product rule to $9 x^{16}$ .

$\log_{2} (9^{\frac{1}{2}} {(x^{16})}^{\frac{1}{2}}) - \frac{2}{3} \cdot \log_{2} (8 x^{3})$

Step 1.3

Rewrite $9$ as $3^{2}$ .

$\log_{2} ({(3^{2})}^{\frac{1}{2}} {(x^{16})}^{\frac{1}{2}}) - \frac{2}{3} \cdot \log_{2} (8 x^{3})$

Step 1.4

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

$\log_{2} (3^{2 (\frac{1}{2})} {(x^{16})}^{\frac{1}{2}}) - \frac{2}{3} \cdot \log_{2} (8 x^{3})$

Step 1.5

Cancel the common factor of $2$ .

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Step 1.5.1

Cancel the common factor.

$\log_{2} (3^{2 (\frac{1}{2})} {(x^{16})}^{\frac{1}{2}}) - \frac{2}{3} \cdot \log_{2} (8 x^{3})$

Step 1.5.2

Rewrite the expression.

$\log_{2} (3^{1} {(x^{16})}^{\frac{1}{2}}) - \frac{2}{3} \cdot \log_{2} (8 x^{3})$

Step 1.6

Evaluate the exponent.

$\log_{2} (3 {(x^{16})}^{\frac{1}{2}}) - \frac{2}{3} \cdot \log_{2} (8 x^{3})$

Step 1.7

Multiply the exponents in ${(x^{16})}^{\frac{1}{2}}$ .

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Step 1.7.1

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

$\log_{2} (3 x^{16 (\frac{1}{2})}) - \frac{2}{3} \cdot \log_{2} (8 x^{3})$

Step 1.7.2

Cancel the common factor of $2$ .

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Step 1.7.2.1

Factor $2$ out of $16$ .

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

Step 1.7.2.2

Cancel the common factor.

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

Step 1.7.2.3

Rewrite the expression.

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

Step 1.8

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

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Step 1.8.1

Reorder $\log_{2} (8 x^{3})$ and $\frac{2}{3}$ .

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

Step 1.8.2

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

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

Step 1.9

Apply the product rule to $8 x^{3}$ .

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

Step 1.10

Rewrite $8$ as $2^{3}$ .

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

Step 1.11

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

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

Step 1.12

Cancel the common factor of $3$ .

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Step 1.12.1

Cancel the common factor.

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

Step 1.12.2

Rewrite the expression.

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

Step 1.13

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

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

Step 1.14

Multiply the exponents in ${(x^{3})}^{\frac{2}{3}}$ .

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Step 1.14.1

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

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

Step 1.14.2

Cancel the common factor of $3$ .

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Step 1.14.2.1

Cancel the common factor.

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

Step 1.14.2.2

Rewrite the expression.

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

Step 2

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

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

Step 3

Cancel the common factor of $x^{8}$ and $x^{2}$ .

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Step 3.1

Factor $x^{2}$ out of $3 x^{8}$ .

$\log_{2} (\frac{x^{2} (3 x^{6})}{4 x^{2}})$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $x^{2}$ out of $4 x^{2}$ .

$\log_{2} (\frac{x^{2} (3 x^{6})}{x^{2} \cdot 4})$

Step 3.2.2

Cancel the common factor.

$\log_{2} (\frac{x^{2} (3 x^{6})}{x^{2} \cdot 4})$

Step 3.2.3

Rewrite the expression.

$\log_{2} (\frac{3 x^{6}}{4})$