Algebra Examples

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

Step 1

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

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y^{3} x^{2}}$ as ${(y^{3} x^{2})}^{\frac{1}{2}}$ .

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

Step 3

Expand $\log ({(y^{3} x^{2})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Step 4

Combine $\frac{1}{2}$ and $\log (y^{3} x^{2})$ .

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

Step 5

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

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

Step 6

Expand $\log (y^{3})$ by moving $3$ outside the logarithm.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Combine the numerators over the common denominator.

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

Step 11

Simplify the numerator.

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Step 11.1

Move $2$ to the left of $\log (z)$ .

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

Step 11.2

Apply the distributive property.

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

Step 11.3

Multiply $3$ by $- 1$ .

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

Step 11.4

Multiply $2$ by $- 1$ .

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