Precalculus Examples

$4 (\log_{3} (x) \cdot (4 \log_{3} (y)) - 4 \log_{3} (z))$

Step 1

Simplify each term.

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Step 1.1

Rewrite using the commutative property of multiplication.

$4 (4 \log_{3} (x) \log_{3} (y) - 4 \log_{3} (z))$

Step 1.2

Simplify $4 \log_{3} (x)$ by moving $4$ inside the logarithm.

$4 (\log_{3} (x^{4}) \log_{3} (y) - 4 \log_{3} (z))$

Step 1.3

Simplify $- 4 \log_{3} (z)$ by moving $4$ inside the logarithm.

$4 (\log_{3} (x^{4}) \log_{3} (y) - \log_{3} (z^{4}))$

Step 2

Apply the distributive property.

$4 (\log_{3} (x^{4}) \log_{3} (y)) + 4 (- \log_{3} (z^{4}))$

Step 3

Multiply $4 (\log_{3} (x^{4}) \log_{3} (y))$ .

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Step 3.1

Reorder $\log_{3} (x^{4})$ and $4$ .

$4 \cdot \log_{3} (x^{4}) \log_{3} (y) + 4 (- \log_{3} (z^{4}))$

Step 3.2

Simplify $4 \cdot \log_{3} (x^{4})$ by moving $4$ inside the logarithm.

$\log_{3} ({(x^{4})}^{4}) \log_{3} (y) + 4 (- \log_{3} (z^{4}))$

Step 4

Multiply $4 (- \log_{3} (z^{4}))$ .

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Step 4.1

Multiply $- 1$ by $4$ .

$\log_{3} ({(x^{4})}^{4}) \log_{3} (y) - 4 \log_{3} (z^{4})$

Step 4.2

Simplify $- 4 \log_{3} (z^{4})$ by moving $4$ inside the logarithm.

$\log_{3} ({(x^{4})}^{4}) \log_{3} (y) - \log_{3} ({(z^{4})}^{4})$

Step 5

Simplify each term.

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Step 5.1

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

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Step 5.1.1

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

$\log_{3} (x^{4 \cdot 4}) \log_{3} (y) - \log_{3} ({(z^{4})}^{4})$

Step 5.1.2

Multiply $4$ by $4$ .

$\log_{3} (x^{16}) \log_{3} (y) - \log_{3} ({(z^{4})}^{4})$

Step 5.2

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

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Step 5.2.1

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

$\log_{3} (x^{16}) \log_{3} (y) - \log_{3} (z^{4 \cdot 4})$

Step 5.2.2

Multiply $4$ by $4$ .

$\log_{3} (x^{16}) \log_{3} (y) - \log_{3} (z^{16})$