Algebra Examples

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

Step 1

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

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

Step 2

Simplify each term.

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Step 2.1

Cancel the common factor of $x^{7}$ and $x^{2}$ .

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Step 2.1.1

Factor $x^{2}$ out of $x^{7}$ .

$\log_{4} (\frac{x^{2} x^{5}}{x^{2}}) + 4 \cdot \log_{4} (x^{3})$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Multiply by $1$ .

$\log_{4} (\frac{x^{2} x^{5}}{x^{2} \cdot 1}) + 4 \cdot \log_{4} (x^{3})$

Step 2.1.2.2

Cancel the common factor.

$\log_{4} (\frac{x^{2} x^{5}}{x^{2} \cdot 1}) + 4 \cdot \log_{4} (x^{3})$

Step 2.1.2.3

Rewrite the expression.

$\log_{4} (\frac{x^{5}}{1}) + 4 \cdot \log_{4} (x^{3})$

Step 2.1.2.4

Divide $x^{5}$ by $1$ .

$\log_{4} (x^{5}) + 4 \cdot \log_{4} (x^{3})$

Step 2.2

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

$\log_{4} (x^{5}) + \log_{4} ({(x^{3})}^{4})$

Step 2.3

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

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Step 2.3.1

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

$\log_{4} (x^{5}) + \log_{4} (x^{3 \cdot 4})$

Step 2.3.2

Multiply $3$ by $4$ .

$\log_{4} (x^{5}) + \log_{4} (x^{12})$

Step 3

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

$\log_{4} (x^{5} x^{12})$

Step 4

Multiply $x^{5}$ by $x^{12}$ by adding the exponents.

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Step 4.1

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

$\log_{4} (x^{5 + 12})$

Step 4.2

Add $5$ and $12$ .

$\log_{4} (x^{17})$