Precalculus Examples

$\frac{1}{3} \cdot \log (x^{18}) - 2 \log (3) + 4 \log (x)$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{3} \log (x^{18})$ by moving $\frac{1}{3}$ inside the logarithm.

$\log ({(x^{18})}^{\frac{1}{3}}) - 2 \log (3) + 4 \log (x)$

Step 1.2

Multiply the exponents in ${(x^{18})}^{\frac{1}{3}}$ .

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Step 1.2.1

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

$\log (x^{18 (\frac{1}{3})}) - 2 \log (3) + 4 \log (x)$

Step 1.2.2

Cancel the common factor of $3$ .

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Step 1.2.2.1

Factor $3$ out of $18$ .

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

Step 1.2.2.2

Cancel the common factor.

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

Step 1.2.2.3

Rewrite the expression.

$\log (x^{6}) - 2 \log (3) + 4 \log (x)$

Step 1.3

Simplify $- 2 \log (3)$ by moving $2$ inside the logarithm.

$\log (x^{6}) - \log (3^{2}) + 4 \log (x)$

Step 1.4

Raise $3$ to the power of $2$ .

$\log (x^{6}) - \log (9) + 4 \log (x)$

Step 1.5

Simplify $4 \log (x)$ by moving $4$ inside the logarithm.

$\log (x^{6}) - \log (9) + \log (x^{4})$

Step 2

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

$\log (\frac{x^{6}}{9}) + \log (x^{4})$

Step 3

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

$\log (\frac{x^{6}}{9} x^{4})$

Step 4

Multiply $\frac{x^{6}}{9} x^{4}$ .

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Step 4.1

Combine $\frac{x^{6}}{9}$ and $x^{4}$ .

$\log (\frac{x^{6} x^{4}}{9})$

Step 4.2

Multiply $x^{6}$ by $x^{4}$ by adding the exponents.

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Step 4.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log (\frac{x^{6 + 4}}{9})$

Step 4.2.2

Add $6$ and $4$ .

$\log (\frac{x^{10}}{9})$