Precalculus Examples

$2 \log_{2} (w) + 2 (\log_{2} (z) - 2 \log_{2} (y))$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log_{2} (w)$ by moving $2$ inside the logarithm.

$\log_{2} (w^{2}) + 2 (\log_{2} (z) - 2 \log_{2} (y))$

Step 1.2

Simplify $- 2 \log_{2} (y)$ by moving $2$ inside the logarithm.

$\log_{2} (w^{2}) + 2 (\log_{2} (z) - \log_{2} (y^{2}))$

Step 1.3

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

$\log_{2} (w^{2}) + 2 \log_{2} (\frac{z}{y^{2}})$

Step 1.4

Simplify $2 \log_{2} (\frac{z}{y^{2}})$ by moving $2$ inside the logarithm.

$\log_{2} (w^{2}) + \log_{2} ({(\frac{z}{y^{2}})}^{2})$

Step 1.5

Apply the product rule to $\frac{z}{y^{2}}$ .

$\log_{2} (w^{2}) + \log_{2} (\frac{z^{2}}{{(y^{2})}^{2}})$

Step 1.6

Multiply the exponents in ${(y^{2})}^{2}$ .

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Step 1.6.1

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

$\log_{2} (w^{2}) + \log_{2} (\frac{z^{2}}{y^{2 \cdot 2}})$

Step 1.6.2

Multiply $2$ by $2$ .

$\log_{2} (w^{2}) + \log_{2} (\frac{z^{2}}{y^{4}})$

Step 2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log_{2} (w^{2} \frac{z^{2}}{y^{4}})$

Step 3

Combine $w^{2}$ and $\frac{z^{2}}{y^{4}}$ .

$\log_{2} (\frac{w^{2} z^{2}}{y^{4}})$