Precalculus Examples

$5 \log (x) + \log (3) - (3 \log (3) + 5 \log (y))$

Step 1

Simplify each term.

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Step 1.1

Simplify $5 \log (x)$ by moving $5$ inside the logarithm.

$\log (x^{5}) + \log (3) - (3 \log (3) + 5 \log (y))$

Step 1.2

Simplify each term.

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Step 1.2.1

Simplify $3 \log (3)$ by moving $3$ inside the logarithm.

$\log (x^{5}) + \log (3) - (\log (3^{3}) + 5 \log (y))$

Step 1.2.2

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

$\log (x^{5}) + \log (3) - (\log (27) + 5 \log (y))$

Step 1.2.3

Simplify $5 \log (y)$ by moving $5$ inside the logarithm.

$\log (x^{5}) + \log (3) - (\log (27) + \log (y^{5}))$

Step 1.3

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

$\log (x^{5}) + \log (3) - \log (27 y^{5})$

Step 2

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

$\log (x^{5} \cdot 3) - \log (27 y^{5})$

Step 3

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

$\log (\frac{x^{5} \cdot 3}{27 y^{5}})$

Step 4

Cancel the common factor of $3$ and $27$ .

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Step 4.1

Factor $3$ out of $x^{5} \cdot 3$ .

$\log (\frac{3 \cdot x^{5}}{27 y^{5}})$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $3$ out of $27 y^{5}$ .

$\log (\frac{3 \cdot x^{5}}{3 (9 y^{5})})$

Step 4.2.2

Cancel the common factor.

$\log (\frac{3 \cdot x^{5}}{3 (9 y^{5})})$

Step 4.2.3

Rewrite the expression.

$\log (\frac{x^{5}}{9 y^{5}})$