Precalculus Examples

$\log_{11} (\frac{x^{2} \sqrt{x}}{y})$

Step 1

Rewrite $\log_{11} (\frac{x^{2} \sqrt{x}}{y})$ as $\log_{11} (x^{2} \sqrt{x}) - \log_{11} (y)$ .

$\log_{11} (x^{2} \sqrt{x}) - \log_{11} (y)$

Step 2

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

$\log_{11} (x^{2} x^{\frac{1}{2}}) - \log_{11} (y)$

Step 3

Multiply $x^{2}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 3.1

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

$\log_{11} (x^{2 + \frac{1}{2}}) - \log_{11} (y)$

Step 3.2

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

$\log_{11} (x^{2 \cdot \frac{2}{2} + \frac{1}{2}}) - \log_{11} (y)$

Step 3.3

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

$\log_{11} (x^{\frac{2 \cdot 2}{2} + \frac{1}{2}}) - \log_{11} (y)$

Step 3.4

Combine the numerators over the common denominator.

$\log_{11} (x^{\frac{2 \cdot 2 + 1}{2}}) - \log_{11} (y)$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $2$ by $2$ .

$\log_{11} (x^{\frac{4 + 1}{2}}) - \log_{11} (y)$

Step 3.5.2

Add $4$ and $1$ .

$\log_{11} (x^{\frac{5}{2}}) - \log_{11} (y)$

Step 4

Expand $\log_{11} (x^{\frac{5}{2}})$ by moving $\frac{5}{2}$ outside the logarithm.

$\frac{5}{2} \log_{11} (x) - \log_{11} (y)$

Step 5

Combine $\frac{5}{2}$ and $\log_{11} (x)$ .

$\frac{5 \log_{11} (x)}{2} - \log_{11} (y)$