Precalculus Examples

$\frac{1}{3} \cdot \log_{x} (8) + 2 \log_{x} (6) - \log_{x} (3)$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{3} \log_{x} (8)$ by moving $\frac{1}{3}$ inside the logarithm.

$\log_{x} (8^{\frac{1}{3}}) + 2 \log_{x} (6) - \log_{x} (3)$

Step 1.2

Rewrite $8$ as $2^{3}$ .

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

Step 1.3

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

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

Step 1.4

Cancel the common factor of $3$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Rewrite the expression.

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

Step 1.5

Evaluate the exponent.

$\log_{x} (2) + 2 \log_{x} (6) - \log_{x} (3)$

Step 1.6

Simplify $2 \log_{x} (6)$ by moving $2$ inside the logarithm.

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

Step 1.7

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

$\log_{x} (2) + \log_{x} (36) - \log_{x} (3)$

Step 2

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

$\log_{x} (2 \cdot 36) - \log_{x} (3)$

Step 3

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

$\log_{x} (\frac{2 \cdot 36}{3})$

Step 4

Cancel the common factor of $36$ and $3$ .

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Step 4.1

Factor $3$ out of $2 \cdot 36$ .

$\log_{x} (\frac{3 (2 \cdot 12)}{3})$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $3$ out of $3$ .

$\log_{x} (\frac{3 (2 \cdot 12)}{3 (1)})$

Step 4.2.2

Cancel the common factor.

$\log_{x} (\frac{3 (2 \cdot 12)}{3 \cdot 1})$

Step 4.2.3

Rewrite the expression.

$\log_{x} (\frac{2 \cdot 12}{1})$

Step 4.2.4

Divide $2 \cdot 12$ by $1$ .

$\log_{x} (2 \cdot 12)$

Step 5

Multiply $2$ by $12$ .

$\log_{x} (24)$