Precalculus Examples

$3 \log (x^{\frac{1}{3}}) - \frac{3}{4} \cdot \log (2 y) - 4 \log (3 \sqrt{y})$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \log (x^{\frac{1}{3}})$ by moving $3$ inside the logarithm.

$\log ({(x^{\frac{1}{3}})}^{3}) - \frac{3}{4} \cdot \log (2 y) - 4 \log (3 \sqrt{y})$

Step 1.2

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 1.2.1

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

$\log (x^{\frac{1}{3} \cdot 3}) - \frac{3}{4} \cdot \log (2 y) - 4 \log (3 \sqrt{y})$

Step 1.2.2

Cancel the common factor of $3$ .

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Step 1.2.2.1

Cancel the common factor.

$\log (x^{\frac{1}{3} \cdot 3}) - \frac{3}{4} \cdot \log (2 y) - 4 \log (3 \sqrt{y})$

Step 1.2.2.2

Rewrite the expression.

$\log (x^{1}) - \frac{3}{4} \cdot \log (2 y) - 4 \log (3 \sqrt{y})$

Step 1.3

Simplify.

$\log (x) - \frac{3}{4} \cdot \log (2 y) - 4 \log (3 \sqrt{y})$

Step 1.4

Multiply $- \frac{3}{4} \log (2 y)$ .

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Step 1.4.1

Reorder $\log (2 y)$ and $\frac{3}{4}$ .

$\log (x) - (\frac{3}{4} \log (2 y)) - 4 \log (3 \sqrt{y})$

Step 1.4.2

Simplify $\frac{3}{4} \log (2 y)$ by moving $\frac{3}{4}$ inside the logarithm.

$\log (x) - \log ({(2 y)}^{\frac{3}{4}}) - 4 \log (3 \sqrt{y})$

Step 1.5

Apply the product rule to $2 y$ .

$\log (x) - \log (2^{\frac{3}{4}} y^{\frac{3}{4}}) - 4 \log (3 \sqrt{y})$

Step 1.6

Simplify $- 4 \log (3 \sqrt{y})$ by moving $4$ inside the logarithm.

$\log (x) - \log (2^{\frac{3}{4}} y^{\frac{3}{4}}) - \log ({(3 \sqrt{y})}^{4})$

Step 1.7

Apply the product rule to $3 \sqrt{y}$ .

$\log (x) - \log (2^{\frac{3}{4}} y^{\frac{3}{4}}) - \log (3^{4} {\sqrt{y}}^{4})$

Step 1.8

Raise $3$ to the power of $4$ .

$\log (x) - \log (2^{\frac{3}{4}} y^{\frac{3}{4}}) - \log (81 {\sqrt{y}}^{4})$

Step 1.9

Rewrite ${\sqrt{y}}^{4}$ as $y^{2}$ .

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Step 1.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

$\log (x) - \log (2^{\frac{3}{4}} y^{\frac{3}{4}}) - \log (81 {(y^{\frac{1}{2}})}^{4})$

Step 1.9.2

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

$\log (x) - \log (2^{\frac{3}{4}} y^{\frac{3}{4}}) - \log (81 y^{\frac{1}{2} \cdot 4})$

Step 1.9.3

Combine $\frac{1}{2}$ and $4$ .

$\log (x) - \log (2^{\frac{3}{4}} y^{\frac{3}{4}}) - \log (81 y^{\frac{4}{2}})$

Step 1.9.4

Cancel the common factor of $4$ and $2$ .

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Step 1.9.4.1

Factor $2$ out of $4$ .

$\log (x) - \log (2^{\frac{3}{4}} y^{\frac{3}{4}}) - \log (81 y^{\frac{2 \cdot 2}{2}})$

Step 1.9.4.2

Cancel the common factors.

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Step 1.9.4.2.1

Factor $2$ out of $2$ .

$\log (x) - \log (2^{\frac{3}{4}} y^{\frac{3}{4}}) - \log (81 y^{\frac{2 \cdot 2}{2 (1)}})$

Step 1.9.4.2.2

Cancel the common factor.

$\log (x) - \log (2^{\frac{3}{4}} y^{\frac{3}{4}}) - \log (81 y^{\frac{2 \cdot 2}{2 \cdot 1}})$

Step 1.9.4.2.3

Rewrite the expression.

$\log (x) - \log (2^{\frac{3}{4}} y^{\frac{3}{4}}) - \log (81 y^{\frac{2}{1}})$

Step 1.9.4.2.4

Divide $2$ by $1$ .

$\log (x) - \log (2^{\frac{3}{4}} y^{\frac{3}{4}}) - \log (81 y^{2})$

Step 2

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

$\log (\frac{x}{2^{\frac{3}{4}} y^{\frac{3}{4}}}) - \log (81 y^{2})$

Step 3

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

$\log (\frac{\frac{x}{2^{\frac{3}{4}} y^{\frac{3}{4}}}}{81 y^{2}})$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{x}{2^{\frac{3}{4}} y^{\frac{3}{4}}} \cdot \frac{1}{81 y^{2}})$

Step 5

Combine.

$\log (\frac{x \cdot 1}{2^{\frac{3}{4}} y^{\frac{3}{4}} (81 y^{2})})$

Step 6

Multiply $y^{\frac{3}{4}}$ by $y^{2}$ by adding the exponents.

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Step 6.1

Move $y^{2}$ .

$\log (\frac{x \cdot 1}{2^{\frac{3}{4}} (y^{2} y^{\frac{3}{4}}) \cdot 81})$

Step 6.2

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

$\log (\frac{x \cdot 1}{2^{\frac{3}{4}} y^{2 + \frac{3}{4}} \cdot 81})$

Step 6.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\log (\frac{x \cdot 1}{2^{\frac{3}{4}} y^{2 \cdot \frac{4}{4} + \frac{3}{4}} \cdot 81})$

Step 6.4

Combine $2$ and $\frac{4}{4}$ .

$\log (\frac{x \cdot 1}{2^{\frac{3}{4}} y^{\frac{2 \cdot 4}{4} + \frac{3}{4}} \cdot 81})$

Step 6.5

Combine the numerators over the common denominator.

$\log (\frac{x \cdot 1}{2^{\frac{3}{4}} y^{\frac{2 \cdot 4 + 3}{4}} \cdot 81})$

Step 6.6

Simplify the numerator.

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Step 6.6.1

Multiply $2$ by $4$ .

$\log (\frac{x \cdot 1}{2^{\frac{3}{4}} y^{\frac{8 + 3}{4}} \cdot 81})$

Step 6.6.2

Add $8$ and $3$ .

$\log (\frac{x \cdot 1}{2^{\frac{3}{4}} y^{\frac{11}{4}} \cdot 81})$

Step 7

Multiply $x$ by $1$ .

$\log (\frac{x}{2^{\frac{3}{4}} y^{\frac{11}{4}} \cdot 81})$

Step 8

Move $81$ to the left of $2^{\frac{3}{4}} y^{\frac{11}{4}}$ .

$\log (\frac{x}{81 \cdot 2^{\frac{3}{4}} y^{\frac{11}{4}}})$