Precalculus Examples

$2 (\log_{2} ({(x - 1)}^{3}) + 7 \log_{2} ({(x + 4)}^{5}))$

Step 1

Simplify each term.

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Step 1.1

Simplify $7 \log_{2} ({(x + 4)}^{5})$ by moving $7$ inside the logarithm.

$2 (\log_{2} ({(x - 1)}^{3}) + \log_{2} ({({(x + 4)}^{5})}^{7}))$

Step 1.2

Multiply the exponents in ${({(x + 4)}^{5})}^{7}$ .

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Step 1.2.1

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

$2 (\log_{2} ({(x - 1)}^{3}) + \log_{2} ({(x + 4)}^{5 \cdot 7}))$

Step 1.2.2

Multiply $5$ by $7$ .

$2 (\log_{2} ({(x - 1)}^{3}) + \log_{2} ({(x + 4)}^{35}))$

Step 2

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

$2 \log_{2} ({(x - 1)}^{3} {(x + 4)}^{35})$

Step 3

Simplify $2 \log_{2} ({(x - 1)}^{3} {(x + 4)}^{35})$ by moving $2$ inside the logarithm.

$\log_{2} ({({(x - 1)}^{3} {(x + 4)}^{35})}^{2})$

Step 4

Apply the product rule to ${(x - 1)}^{3} {(x + 4)}^{35}$ .

$\log_{2} ({({(x - 1)}^{3})}^{2} {({(x + 4)}^{35})}^{2})$

Step 5

Multiply the exponents in ${({(x - 1)}^{3})}^{2}$ .

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Step 5.1

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

$\log_{2} ({(x - 1)}^{3 \cdot 2} {({(x + 4)}^{35})}^{2})$

Step 5.2

Multiply $3$ by $2$ .

$\log_{2} ({(x - 1)}^{6} {({(x + 4)}^{35})}^{2})$

Step 6

Multiply the exponents in ${({(x + 4)}^{35})}^{2}$ .

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Step 6.1

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

$\log_{2} ({(x - 1)}^{6} {(x + 4)}^{35 \cdot 2})$

Step 6.2

Multiply $35$ by $2$ .

$\log_{2} ({(x - 1)}^{6} {(x + 4)}^{70})$