Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \log_{5} (2)$ by moving $3$ inside the logarithm.

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

Step 1.2

Raise $2$ to the power of $3$ .

$\log_{5} (8) + 5 \log_{5} (x) - \frac{1}{5} \cdot \log_{5} (y)$

Step 1.3

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

$\log_{5} (8) + \log_{5} (x^{5}) - \frac{1}{5} \cdot \log_{5} (y)$

Step 1.4

Multiply $- \frac{1}{5} \log_{5} (y)$ .

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Step 1.4.1

Reorder $\log_{5} (y)$ and $\frac{1}{5}$ .

$\log_{5} (8) + \log_{5} (x^{5}) - (\frac{1}{5} \log_{5} (y))$

Step 1.4.2

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

$\log_{5} (8) + \log_{5} (x^{5}) - \log_{5} (y^{\frac{1}{5}})$

Step 2

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

$\log_{5} (8 x^{5}) - \log_{5} (y^{\frac{1}{5}})$

Step 3

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

$\log_{5} (\frac{8 x^{5}}{y^{\frac{1}{5}}})$