Precalculus Examples

$\log (4 x^{2}) - \log (y^{\frac{1}{2}} z^{9})$

Step 1

Rewrite $\log (4 x^{2})$ as $\log (4) + \log (x^{2})$ .

$\log (4) + \log (x^{2}) - \log (y^{\frac{1}{2}} z^{9})$

Step 2

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

$\log (4) + 2 \log (x) - \log (y^{\frac{1}{2}} z^{9})$

Step 3

Rewrite $\log (y^{\frac{1}{2}} z^{9})$ as $\log (y^{\frac{1}{2}}) + \log (z^{9})$ .

$\log (4) + 2 \log (x) - (\log (y^{\frac{1}{2}}) + \log (z^{9}))$

Step 4

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

$\log (4) + 2 \log (x) - (\frac{1}{2} \log (y) + \log (z^{9}))$

Step 5

Expand $\log (z^{9})$ by moving $9$ outside the logarithm.

$\log (4) + 2 \log (x) - (\frac{1}{2} \log (y) + 9 \log (z))$

Step 6

Simplify each term.

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Step 6.1

Rewrite $\log (4)$ as $\log (2^{2})$ .

$\log (2^{2}) + 2 \log (x) - (\frac{1}{2} \log (y) + 9 \log (z))$

Step 6.2

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

$2 \log (2) + 2 \log (x) - (\frac{1}{2} \log (y) + 9 \log (z))$

Step 6.3

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

$2 \log (2) + 2 \log (x) - (\frac{\log (y)}{2} + 9 \log (z))$

Step 6.4

Apply the distributive property.

$2 \log (2) + 2 \log (x) - \frac{\log (y)}{2} - (9 \log (z))$

Step 6.5

Multiply $9$ by $- 1$ .

$2 \log (2) + 2 \log (x) - \frac{\log (y)}{2} - 9 \log (z)$