Precalculus Examples

$\frac{2}{3} \cdot \log_{3} (x^{3}) - \log_{3} (\frac{3}{4} y^{4}) - \log_{3} (10)$

Step 1

Simplify each term.

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Step 1.1

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

$\log_{3} ({(x^{3})}^{\frac{2}{3}}) - \log_{3} (\frac{3}{4} y^{4}) - \log_{3} (10)$

Step 1.2

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

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Step 1.2.1

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

$\log_{3} (x^{3 (\frac{2}{3})}) - \log_{3} (\frac{3}{4} y^{4}) - \log_{3} (10)$

Step 1.2.2

Cancel the common factor of $3$ .

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Step 1.2.2.1

Cancel the common factor.

$\log_{3} (x^{3 (\frac{2}{3})}) - \log_{3} (\frac{3}{4} y^{4}) - \log_{3} (10)$

Step 1.2.2.2

Rewrite the expression.

$\log_{3} (x^{2}) - \log_{3} (\frac{3}{4} y^{4}) - \log_{3} (10)$

Step 1.3

Combine $\frac{3}{4}$ and $y^{4}$ .

$\log_{3} (x^{2}) - \log_{3} (\frac{3 y^{4}}{4}) - \log_{3} (10)$

Step 2

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

$\log_{3} (\frac{x^{2}}{\frac{3 y^{4}}{4}}) - \log_{3} (10)$

Step 3

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

$\log_{3} (\frac{\frac{x^{2}}{\frac{3 y^{4}}{4}}}{10})$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\log_{3} (\frac{x^{2}}{\frac{3 y^{4}}{4}} \cdot \frac{1}{10})$

Step 5

Combine.

$\log_{3} (\frac{x^{2} \cdot 1}{\frac{3 y^{4}}{4} \cdot 10})$

Step 6

Multiply $x^{2}$ by $1$ .

$\log_{3} (\frac{x^{2}}{\frac{3 y^{4}}{4} \cdot 10})$

Step 7

Combine $\frac{3 y^{4}}{4}$ and $10$ .

$\log_{3} (\frac{x^{2}}{\frac{3 y^{4} \cdot 10}{4}})$

Step 8

Multiply $10$ by $3$ .

$\log_{3} (\frac{x^{2}}{\frac{30 y^{4}}{4}})$

Step 9

Reduce the expression $\frac{30 y^{4}}{4}$ by cancelling the common factors.

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Step 9.1

Factor $2$ out of $30 y^{4}$ .

$\log_{3} (\frac{x^{2}}{\frac{2 (15 y^{4})}{4}})$

Step 9.2

Factor $2$ out of $4$ .

$\log_{3} (\frac{x^{2}}{\frac{2 (15 y^{4})}{2 (2)}})$

Step 9.3

Cancel the common factor.

$\log_{3} (\frac{x^{2}}{\frac{2 (15 y^{4})}{2 \cdot 2}})$

Step 9.4

Rewrite the expression.

$\log_{3} (\frac{x^{2}}{\frac{15 y^{4}}{2}})$

Step 10

Multiply the numerator by the reciprocal of the denominator.

$\log_{3} (x^{2} \frac{2}{15 y^{4}})$

Step 11

Combine $x^{2}$ and $\frac{2}{15 y^{4}}$ .

$\log_{3} (\frac{x^{2} \cdot 2}{15 y^{4}})$

Step 12

Move $2$ to the left of $x^{2}$ .

$\log_{3} (\frac{2 x^{2}}{15 y^{4}})$