Precalculus Examples

$(4 \log_{3} (x + 6) + 7 \log_{3} (x + 4)) - \frac{1}{2} \cdot \log_{3} (x)$

Step 1

Simplify each term.

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Step 1.1

Simplify $4 \log_{3} (x + 6)$ by moving $4$ inside the logarithm.

$\log_{3} ({(x + 6)}^{4}) + 7 \log_{3} (x + 4) - \frac{1}{2} \cdot \log_{3} (x)$

Step 1.2

Simplify $7 \log_{3} (x + 4)$ by moving $7$ inside the logarithm.

$\log_{3} ({(x + 6)}^{4}) + \log_{3} ({(x + 4)}^{7}) - \frac{1}{2} \cdot \log_{3} (x)$

Step 1.3

Multiply $- \frac{1}{2} \log_{3} (x)$ .

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Step 1.3.1

Reorder $\log_{3} (x)$ and $\frac{1}{2}$ .

$\log_{3} ({(x + 6)}^{4}) + \log_{3} ({(x + 4)}^{7}) - (\frac{1}{2} \log_{3} (x))$

Step 1.3.2

Simplify $\frac{1}{2} \log_{3} (x)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{3} ({(x + 6)}^{4}) + \log_{3} ({(x + 4)}^{7}) - \log_{3} (x^{\frac{1}{2}})$

Step 2

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

$\log_{3} ({(x + 6)}^{4} {(x + 4)}^{7}) - \log_{3} (x^{\frac{1}{2}})$

Step 3

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

$\log_{3} (\frac{{(x + 6)}^{4} {(x + 4)}^{7}}{x^{\frac{1}{2}}})$