Write as a Single Logarithm 1/2* log base 2 of 9x^16-2/3* log base 2 of 8x^3

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

Step 1

Simplify each term.

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

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

Cancel the common factor of $2$ .

Factor $2$ out of $16$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

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 $x^{2}$ .

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Write as a Single Logarithm 1/2* log base 2 of 9x^16-2/3* log base 2 of 8x^3

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