Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.2.2

Multiply $2$ by $3$ .

$\log_{4} (x^{6}) - 4 \log_{4} (x) + 5 \log_{4} (x)$

Step 1.3

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

$\log_{4} (x^{6}) - \log_{4} (x^{4}) + 5 \log_{4} (x)$

Step 1.4

Simplify $5 \log_{4} (x)$ by moving $5$ inside the logarithm.

$\log_{4} (x^{6}) - \log_{4} (x^{4}) + \log_{4} (x^{5})$

Step 2

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

$\log_{4} (\frac{x^{6}}{x^{4}}) + \log_{4} (x^{5})$

Step 3

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

$\log_{4} (\frac{x^{6}}{x^{4}} x^{5})$

Step 4

Cancel the common factor of $x^{4}$ .

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Step 4.1

Factor $x^{4}$ out of $x^{5}$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

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

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Step 5.1

Multiply $x^{6}$ by $x$ .

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Step 5.1.1

Raise $x$ to the power of $1$ .

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

Step 5.1.2

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

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

Step 5.2

Add $6$ and $1$ .

$\log_{4} (x^{7})$