Precalculus Examples

$3 \log_{5} (x) + \frac{1}{2} \cdot \log_{5} (y) - 2 \log_{5} (2)$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \log_{5} (x)$ by moving $3$ inside the logarithm.

$\log_{5} (x^{3}) + \frac{1}{2} \cdot \log_{5} (y) - 2 \log_{5} (2)$

Step 1.2

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

$\log_{5} (x^{3}) + \log_{5} (y^{\frac{1}{2}}) - 2 \log_{5} (2)$

Step 1.3

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

$\log_{5} (x^{3}) + \log_{5} (y^{\frac{1}{2}}) - \log_{5} (2^{2})$

Step 1.4

Raise $2$ to the power of $2$ .

$\log_{5} (x^{3}) + \log_{5} (y^{\frac{1}{2}}) - \log_{5} (4)$

Step 2

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

$\log_{5} (x^{3} y^{\frac{1}{2}}) - \log_{5} (4)$

Step 3

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

$\log_{5} (\frac{x^{3} y^{\frac{1}{2}}}{4})$