Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Apply the product rule to $2 x$ .

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

Step 1.3

Raise $2$ to the power of $4$ .

$\log_{4} (16 x^{4}) - \log_{4} (x^{5}) + \log_{4} (4 x)$

Step 2

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

$\log_{4} (\frac{16 x^{4}}{x^{5}}) + \log_{4} (4 x)$

Step 3

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

$\log_{4} (\frac{16 x^{4}}{x^{5}} (4 x))$

Step 4

Rewrite using the commutative property of multiplication.

$\log_{4} (4 \frac{16 x^{4}}{x^{5}} x)$

Step 5

Cancel the common factor of $x^{4}$ and $x^{5}$ .

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Step 5.1

Factor $x^{4}$ out of $16 x^{4}$ .

$\log_{4} (4 \frac{x^{4} \cdot 16}{x^{5}} x)$

Step 5.2

Cancel the common factors.

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Step 5.2.1

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

$\log_{4} (4 \frac{x^{4} \cdot 16}{x^{4} x} x)$

Step 5.2.2

Cancel the common factor.

$\log_{4} (4 \frac{x^{4} \cdot 16}{x^{4} x} x)$

Step 5.2.3

Rewrite the expression.

$\log_{4} (4 \frac{16}{x} x)$

Step 6

Multiply $4 \frac{16}{x}$ .

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Step 6.1

Combine $4$ and $\frac{16}{x}$ .

$\log_{4} (\frac{4 \cdot 16}{x} x)$

Step 6.2

Multiply $4$ by $16$ .

$\log_{4} (\frac{64}{x} x)$

Step 7

Cancel the common factor of $x$ .

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Step 7.1

Cancel the common factor.

$\log_{4} (\frac{64}{x} x)$

Step 7.2

Rewrite the expression.

$\log_{4} (64)$

Step 8

Logarithm base $4$ of $64$ is $3$ .

$3$