Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{2} \log_{2} (x + 1)$ by moving $\frac{1}{2}$ inside the logarithm.

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

Step 1.2

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

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

Step 2

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

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

Step 3

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

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

Step 4

Combine $\frac{{(x + 1)}^{\frac{1}{2}}}{y}$ and $z^{2}$ .

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

Step 5

Reorder factors in $\log_{2} (\frac{{(x + 1)}^{\frac{1}{2}} z^{2}}{y})$ .

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