Precalculus Examples

$\log (x) + \log (x^{2} y) + 3 \log (y)$

Step 1

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

$\log (x (x^{2} y)) + 3 \log (y)$

Step 2

Simplify each term.

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Step 2.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 2.1.1

Move $x^{2}$ .

$\log (x^{2} x y) + 3 \log (y)$

Step 2.1.2

Multiply $x^{2}$ by $x$ .

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Step 2.1.2.1

Raise $x$ to the power of $1$ .

$\log (x^{2} x^{1} y) + 3 \log (y)$

Step 2.1.2.2

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

$\log (x^{2 + 1} y) + 3 \log (y)$

Step 2.1.3

Add $2$ and $1$ .

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

Step 2.2

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

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

Step 3

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

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

Step 4

Multiply $y$ by $y^{3}$ by adding the exponents.

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Step 4.1

Move $y^{3}$ .

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

Step 4.2

Multiply $y^{3}$ by $y$ .

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Step 4.2.1

Raise $y$ to the power of $1$ .

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

Step 4.2.2

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

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

Step 4.3

Add $3$ and $1$ .

$\log (x^{3} y^{4})$