Precalculus Examples

$4 (\log (x) + \log (y) + 3 \log (z))$

Step 1

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

$4 (\log (x y) + 3 \log (z))$

Step 2

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

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

Step 3

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

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

Step 4

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

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

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 $x y z^{3}$ .

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

Step 5.2

Apply the product rule to $x y$ .

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Multiply $3$ by $4$ .

$\log (x^{4} y^{4} z^{12})$