Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Multiply $2$ by $3$ .

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

Step 1.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.4

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

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

Step 1.5

Apply the product rule to $\frac{1}{x^{3}}$ .

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

Step 1.6

One to any power is one.

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

Step 1.7

Multiply the exponents in ${(x^{3})}^{4}$ .

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Step 1.7.1

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

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

Step 1.7.2

Multiply $3$ by $4$ .

$\log_{4} (x^{6}) - \log_{4} (\frac{1}{x^{12}}) + 2 \cdot \log_{4} (x)$

Step 1.8

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

$\log_{4} (x^{6}) - \log_{4} (\frac{1}{x^{12}}) + \log_{4} (x^{2})$

Step 2

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

$\log_{4} (\frac{x^{6}}{\frac{1}{x^{12}}}) + \log_{4} (x^{2})$

Step 3

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

$\log_{4} (\frac{x^{6}}{\frac{1}{x^{12}}} x^{2})$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\log_{4} (x^{6} x^{12} x^{2})$

Step 5

Multiply $x^{6}$ by $x^{12}$ by adding the exponents.

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Step 5.1

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

$\log_{4} (x^{6 + 12} x^{2})$

Step 5.2

Add $6$ and $12$ .

$\log_{4} (x^{18} x^{2})$

Step 6

Multiply $x^{18}$ by $x^{2}$ by adding the exponents.

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Step 6.1

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

$\log_{4} (x^{18 + 2})$

Step 6.2

Add $18$ and $2$ .

$\log_{4} (x^{20})$