Precalculus Examples

$4 (\log_{7} (x) + 2 \log_{7} (y) + 3 \log_{7} (z))$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log_{7} (y)$ by moving $2$ inside the logarithm.

$4 (\log_{7} (x) + \log_{7} (y^{2}) + 3 \log_{7} (z))$

Step 1.2

Simplify $3 \log_{7} (z)$ by moving $3$ inside the logarithm.

$4 (\log_{7} (x) + \log_{7} (y^{2}) + \log_{7} (z^{3}))$

Step 2

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

$4 (\log_{7} (x y^{2}) + \log_{7} (z^{3}))$

Step 3

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

$4 \log_{7} (x y^{2} z^{3})$

Step 4

Simplify $4 \log_{7} (x y^{2} z^{3})$ by moving $4$ inside the logarithm.

$\log_{7} ({(x y^{2} z^{3})}^{4})$

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 $x y^{2} z^{3}$ .

$\log_{7} ({(x y^{2})}^{4} {(z^{3})}^{4})$

Step 5.2

Apply the product rule to $x y^{2}$ .

$\log_{7} (x^{4} {(y^{2})}^{4} {(z^{3})}^{4})$

Step 6

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

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Step 6.1

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

$\log_{7} (x^{4} y^{2 \cdot 4} {(z^{3})}^{4})$

Step 6.2

Multiply $2$ by $4$ .

$\log_{7} (x^{4} y^{8} {(z^{3})}^{4})$

Step 7

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

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Step 7.1

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

$\log_{7} (x^{4} y^{8} z^{3 \cdot 4})$

Step 7.2

Multiply $3$ by $4$ .

$\log_{7} (x^{4} y^{8} z^{12})$