Precalculus Examples

$\log (\frac{5 x^{2}}{3 \sqrt{y}})$

Step 1

Rewrite $\log (\frac{5 x^{2}}{3 \sqrt{y}})$ as $\log (5 x^{2}) - \log (3 \sqrt{y})$ .

$\log (5 x^{2}) - \log (3 \sqrt{y})$

Step 2

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

$\log (5 x^{2}) - \log (3 y^{\frac{1}{2}})$

Step 3

Rewrite $\log (5 x^{2})$ as $\log (5) + \log (x^{2})$ .

$\log (5) + \log (x^{2}) - \log (3 y^{\frac{1}{2}})$

Step 4

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

$\log (5) + 2 \log (x) - \log (3 y^{\frac{1}{2}})$

Step 5

Rewrite $\log (3 y^{\frac{1}{2}})$ as $\log (3) + \log (y^{\frac{1}{2}})$ .

$\log (5) + 2 \log (x) - (\log (3) + \log (y^{\frac{1}{2}}))$

Step 6

Expand $\log (y^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log (5) + 2 \log (x) - (\log (3) + \frac{1}{2} \log (y))$

Step 7

Simplify each term.

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Step 7.1

Combine $\frac{1}{2}$ and $\log (y)$ .

$\log (5) + 2 \log (x) - (\log (3) + \frac{\log (y)}{2})$

Step 7.2

Apply the distributive property.

$\log (5) + 2 \log (x) - \log (3) - \frac{\log (y)}{2}$