Precalculus Examples

$3 (\log_{6} (x) + 3 \log_{6} (y) - 4 \log_{6} (z))$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \log_{6} (y)$ by moving $3$ inside the logarithm.

$3 (\log_{6} (x) + \log_{6} (y^{3}) - 4 \log_{6} (z))$

Step 1.2

Simplify $- 4 \log_{6} (z)$ by moving $4$ inside the logarithm.

$3 (\log_{6} (x) + \log_{6} (y^{3}) - \log_{6} (z^{4}))$

Step 2

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

$3 (\log_{6} (x y^{3}) - \log_{6} (z^{4}))$

Step 3

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

$3 \log_{6} (\frac{x y^{3}}{z^{4}})$

Step 4

Simplify $3 \log_{6} (\frac{x y^{3}}{z^{4}})$ by moving $3$ inside the logarithm.

$\log_{6} ({(\frac{x y^{3}}{z^{4}})}^{3})$

Step 5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1

Apply the product rule to $\frac{x y^{3}}{z^{4}}$ .

$\log_{6} (\frac{{(x y^{3})}^{3}}{{(z^{4})}^{3}})$

Step 5.2

Apply the product rule to $x y^{3}$ .

$\log_{6} (\frac{x^{3} {(y^{3})}^{3}}{{(z^{4})}^{3}})$

Step 6

Multiply the exponents in ${(y^{3})}^{3}$ .

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Step 6.1

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

$\log_{6} (\frac{x^{3} y^{3 \cdot 3}}{{(z^{4})}^{3}})$

Step 6.2

Multiply $3$ by $3$ .

$\log_{6} (\frac{x^{3} y^{9}}{{(z^{4})}^{3}})$

Step 7

Multiply the exponents in ${(z^{4})}^{3}$ .

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Step 7.1

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

$\log_{6} (\frac{x^{3} y^{9}}{z^{4 \cdot 3}})$

Step 7.2

Multiply $4$ by $3$ .

$\log_{6} (\frac{x^{3} y^{9}}{z^{12}})$