Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log (x^{5} y^{4})$ by moving $2$ inside the logarithm.

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

Step 1.2

Apply the product rule to $x^{5} y^{4}$ .

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

Step 1.3

Multiply the exponents in ${(x^{5})}^{2}$ .

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Step 1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.3.2

Multiply $5$ by $2$ .

$\log (x^{10} {(y^{4})}^{2}) + 4 \log (\frac{x}{1}) - 5 \log (y)$

Step 1.4

Multiply the exponents in ${(y^{4})}^{2}$ .

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Step 1.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log (x^{10} y^{4 \cdot 2}) + 4 \log (\frac{x}{1}) - 5 \log (y)$

Step 1.4.2

Multiply $4$ by $2$ .

$\log (x^{10} y^{8}) + 4 \log (\frac{x}{1}) - 5 \log (y)$

Step 1.5

Divide $x$ by $1$ .

$\log (x^{10} y^{8}) + 4 \log (x) - 5 \log (y)$

Step 1.6

Simplify $4 \log (x)$ by moving $4$ inside the logarithm.

$\log (x^{10} y^{8}) + \log (x^{4}) - 5 \log (y)$

Step 1.7

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

$\log (x^{10} y^{8}) + \log (x^{4}) - \log (y^{5})$

Step 2

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

$\log (x^{10} y^{8} x^{4}) - \log (y^{5})$

Step 3

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

$\log (\frac{x^{10} y^{8} x^{4}}{y^{5}})$

Step 4

Multiply $x^{10}$ by $x^{4}$ by adding the exponents.

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Step 4.1

Move $x^{4}$ .

$\log (\frac{x^{4} x^{10} y^{8}}{y^{5}})$

Step 4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log (\frac{x^{4 + 10} y^{8}}{y^{5}})$

Step 4.3

Add $4$ and $10$ .

$\log (\frac{x^{14} y^{8}}{y^{5}})$

Step 5

Cancel the common factor of $y^{8}$ and $y^{5}$ .

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Step 5.1

Factor $y^{5}$ out of $x^{14} y^{8}$ .

$\log (\frac{y^{5} (x^{14} y^{3})}{y^{5}})$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Multiply by $1$ .

$\log (\frac{y^{5} (x^{14} y^{3})}{y^{5} \cdot 1})$

Step 5.2.2

Cancel the common factor.

$\log (\frac{y^{5} (x^{14} y^{3})}{y^{5} \cdot 1})$

Step 5.2.3

Rewrite the expression.

$\log (\frac{x^{14} y^{3}}{1})$

Step 5.2.4

Divide $x^{14} y^{3}$ by $1$ .

$\log (x^{14} y^{3})$