Precalculus Examples

$\frac{1}{6} \cdot \log (x^{5} y^{7}) - \frac{6}{7 \log (x^{2 y})}$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{6} \log (x^{5} y^{7})$ by moving $\frac{1}{6}$ inside the logarithm.

$\log ({(x^{5} y^{7})}^{\frac{1}{6}}) - \frac{6}{7 \log (x^{2 y})}$

Step 1.2

Apply the product rule to $x^{5} y^{7}$ .

$\log ({(x^{5})}^{\frac{1}{6}} {(y^{7})}^{\frac{1}{6}}) - \frac{6}{7 \log (x^{2 y})}$

Step 1.3

Multiply the exponents in ${(x^{5})}^{\frac{1}{6}}$ .

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Step 1.3.1

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

$\log (x^{5 (\frac{1}{6})} {(y^{7})}^{\frac{1}{6}}) - \frac{6}{7 \log (x^{2 y})}$

Step 1.3.2

Combine $5$ and $\frac{1}{6}$ .

$\log (x^{\frac{5}{6}} {(y^{7})}^{\frac{1}{6}}) - \frac{6}{7 \log (x^{2 y})}$

Step 1.4

Multiply the exponents in ${(y^{7})}^{\frac{1}{6}}$ .

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Step 1.4.1

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

$\log (x^{\frac{5}{6}} y^{7 (\frac{1}{6})}) - \frac{6}{7 \log (x^{2 y})}$

Step 1.4.2

Combine $7$ and $\frac{1}{6}$ .

$\log (x^{\frac{5}{6}} y^{\frac{7}{6}}) - \frac{6}{7 \log (x^{2 y})}$

Step 1.5

Expand $\log (x^{2 y})$ by moving $2 y$ outside the logarithm.

$\log (x^{\frac{5}{6}} y^{\frac{7}{6}}) - \frac{6}{7 (2 y \log (x))}$

Step 1.6

Cancel the common factor of $6$ and $2$ .

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Step 1.6.1

Factor $2$ out of $6$ .

$\log (x^{\frac{5}{6}} y^{\frac{7}{6}}) - \frac{2 \cdot 3}{7 (2 y \log (x))}$

Step 1.6.2

Cancel the common factors.

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Step 1.6.2.1

Factor $2$ out of $7 (2 y \log (x))$ .

$\log (x^{\frac{5}{6}} y^{\frac{7}{6}}) - \frac{2 \cdot 3}{2 (7 (y \log (x)))}$

Step 1.6.2.2

Cancel the common factor.

$\log (x^{\frac{5}{6}} y^{\frac{7}{6}}) - \frac{2 \cdot 3}{2 (7 (y \log (x)))}$

Step 1.6.2.3

Rewrite the expression.

$\log (x^{\frac{5}{6}} y^{\frac{7}{6}}) - \frac{3}{7 (y \log (x))}$

Step 1.7

Simplify $7 \log (x)$ by moving $7$ inside the logarithm.

$\log (x^{\frac{5}{6}} y^{\frac{7}{6}}) - \frac{3}{y \log (x^{7})}$

Step 2

To write $\log (x^{\frac{5}{6}} y^{\frac{7}{6}})$ as a fraction with a common denominator, multiply by $\frac{y \log (x^{7})}{y \log (x^{7})}$ .

$\log (x^{\frac{5}{6}} y^{\frac{7}{6}}) \cdot \frac{y \log (x^{7})}{y \log (x^{7})} - \frac{3}{y \log (x^{7})}$

Step 3

Combine $\log (x^{\frac{5}{6}} y^{\frac{7}{6}})$ and $\frac{y \log (x^{7})}{y \log (x^{7})}$ .

$\frac{\log (x^{\frac{5}{6}} y^{\frac{7}{6}}) (y \log (x^{7}))}{y \log (x^{7})} - \frac{3}{y \log (x^{7})}$

Step 4

Combine the numerators over the common denominator.

$\frac{\log (x^{\frac{5}{6}} y^{\frac{7}{6}}) (y \log (x^{7})) - 3}{y \log (x^{7})}$

Step 5

Reorder factors in $\frac{\log (x^{\frac{5}{6}} y^{\frac{7}{6}}) (y \log (x^{7})) - 3}{y \log (x^{7})}$ .

$\frac{y \log (x^{\frac{5}{6}} y^{\frac{7}{6}}) \log (x^{7}) - 3}{y \log (x^{7})}$