Precalculus Examples

$3 \log (6) x - \frac{1}{2} \cdot (\log (6) (x^{2} - 2))$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \log (6)$ by moving $3$ inside the logarithm.

$\log (6^{3}) x - \frac{1}{2} \cdot (\log (6) (x^{2} - 2))$

Step 1.2

Raise $6$ to the power of $3$ .

$\log (216) x - \frac{1}{2} \cdot (\log (6) (x^{2} - 2))$

Step 1.3

Apply the distributive property.

$\log (216) x - \frac{1}{2} \cdot (\log (6) x^{2} + \log (6) \cdot - 2)$

Step 1.4

Multiply $\log (6) \cdot - 2$ .

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Step 1.4.1

Reorder $\log (6)$ and $- 2$ .

$\log (216) x - \frac{1}{2} \cdot (\log (6) x^{2} - 2 \cdot \log (6))$

Step 1.4.2

Simplify $- 2 \cdot \log (6)$ by moving $2$ inside the logarithm.

$\log (216) x - \frac{1}{2} \cdot (\log (6) x^{2} - \log (6^{2}))$

Step 1.5

Raise $6$ to the power of $2$ .

$\log (216) x - \frac{1}{2} \cdot (\log (6) x^{2} - \log (36))$

Step 1.6

Apply the distributive property.

$\log (216) x - \frac{1}{2} (\log (6) x^{2}) - \frac{1}{2} (- \log (36))$

Step 1.7

Multiply $- \frac{1}{2} (\log (6) x^{2})$ .

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Step 1.7.1

Reorder $\log (6)$ and $\frac{1}{2}$ .

$\log (216) x - (\frac{1}{2} \log (6)) x^{2} - \frac{1}{2} (- \log (36))$

Step 1.7.2

Simplify $\frac{1}{2} \log (6)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log (216) x - \log (6^{\frac{1}{2}}) x^{2} - \frac{1}{2} (- \log (36))$

Step 1.8

Multiply $- \frac{1}{2} (- \log (36))$ .

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Step 1.8.1

Multiply $- 1$ by $- 1$ .

$\log (216) x - \log (6^{\frac{1}{2}}) x^{2} + 1 (\frac{1}{2}) \log (36)$

Step 1.8.2

Multiply $\frac{1}{2}$ by $1$ .

$\log (216) x - \log (6^{\frac{1}{2}}) x^{2} + \frac{1}{2} \log (36)$

Step 1.8.3

Simplify $\frac{1}{2} \log (36)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log (216) x - \log (6^{\frac{1}{2}}) x^{2} + \log (36^{\frac{1}{2}})$

Step 1.9

Simplify each term.

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Step 1.9.1

Rewrite $36$ as $6^{2}$ .

$\log (216) x - \log (6^{\frac{1}{2}}) x^{2} + \log ({(6^{2})}^{\frac{1}{2}})$

Step 1.9.2

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

$\log (216) x - \log (6^{\frac{1}{2}}) x^{2} + \log (6^{2 (\frac{1}{2})})$

Step 1.9.3

Cancel the common factor of $2$ .

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Step 1.9.3.1

Cancel the common factor.

$\log (216) x - \log (6^{\frac{1}{2}}) x^{2} + \log (6^{2 (\frac{1}{2})})$

Step 1.9.3.2

Rewrite the expression.

$\log (216) x - \log (6^{\frac{1}{2}}) x^{2} + \log (6^{1})$

Step 1.9.4

Evaluate the exponent.

$\log (216) x - \log (6^{\frac{1}{2}}) x^{2} + \log (6)$

Step 2

Reorder factors in $\log (216) x - \log (6^{\frac{1}{2}}) x^{2} + \log (6)$ .

$x \log (216) - x^{2} \log (6^{\frac{1}{2}}) + \log (6)$