Precalculus Examples

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

Step 1

Rewrite $\log_{5} (\frac{x^{2}}{y^{2} z^{3}})$ as $\log_{5} (x^{2}) - \log_{5} (y^{2} z^{3})$ .

$\log_{5} (x^{2}) - \log_{5} (y^{2} z^{3})$

Step 2

Expand $\log_{5} (x^{2})$ by moving $2$ outside the logarithm.

$2 \log_{5} (x) - \log_{5} (y^{2} z^{3})$

Step 3

Rewrite $\log_{5} (y^{2} z^{3})$ as $\log_{5} (y^{2}) + \log_{5} (z^{3})$ .

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

Step 4

Expand $\log_{5} (y^{2})$ by moving $2$ outside the logarithm.

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

Step 5

Expand $\log_{5} (z^{3})$ by moving $3$ outside the logarithm.

$2 \log_{5} (x) - (2 \log_{5} (y) + 3 \log_{5} (z))$

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

$2 \log_{5} (x) - (2 \log_{5} (y)) - (3 \log_{5} (z))$

Step 6.2

Multiply $2$ by $- 1$ .

$2 \log_{5} (x) - 2 \log_{5} (y) - (3 \log_{5} (z))$

Step 6.3

Multiply $3$ by $- 1$ .

$2 \log_{5} (x) - 2 \log_{5} (y) - 3 \log_{5} (z)$