Precalculus Examples

$\frac{1}{2} \cdot (\log_{5} (x + 1) + \log_{5} (y)) + 2 \log_{5} (x + 1)$

Step 1

Simplify each term.

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Step 1.1

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

$\frac{1}{2} \cdot \log_{5} ((x + 1) y) + 2 \log_{5} (x + 1)$

Step 1.2

Apply the distributive property.

$\frac{1}{2} \cdot \log_{5} (x y + 1 y) + 2 \log_{5} (x + 1)$

Step 1.3

Multiply $y$ by $1$ .

$\frac{1}{2} \cdot \log_{5} (x y + y) + 2 \log_{5} (x + 1)$

Step 1.4

Simplify $\frac{1}{2} \log_{5} (x y + y)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{5} ({(x y + y)}^{\frac{1}{2}}) + 2 \log_{5} (x + 1)$

Step 1.5

Simplify $2 \log_{5} (x + 1)$ by moving $2$ inside the logarithm.

$\log_{5} ({(x y + y)}^{\frac{1}{2}}) + \log_{5} ({(x + 1)}^{2})$

Step 2

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

$\log_{5} ({(x y + y)}^{\frac{1}{2}} {(x + 1)}^{2})$