Precalculus Examples

$3 \log (4 x^{3}) - \frac{1}{9} \cdot \log (4 x + 19)$

Step 1

Simplify each term.

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Step 1.1

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

$\log ({(4 x^{3})}^{3}) - \frac{1}{9} \cdot \log (4 x + 19)$

Step 1.2

Apply the product rule to $4 x^{3}$ .

$\log (4^{3} {(x^{3})}^{3}) - \frac{1}{9} \cdot \log (4 x + 19)$

Step 1.3

Raise $4$ to the power of $3$ .

$\log (64 {(x^{3})}^{3}) - \frac{1}{9} \cdot \log (4 x + 19)$

Step 1.4

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

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Step 1.4.1

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

$\log (64 x^{3 \cdot 3}) - \frac{1}{9} \cdot \log (4 x + 19)$

Step 1.4.2

Multiply $3$ by $3$ .

$\log (64 x^{9}) - \frac{1}{9} \cdot \log (4 x + 19)$

Step 1.5

Multiply $- \frac{1}{9} \log (4 x + 19)$ .

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Step 1.5.1

Reorder $\log (4 x + 19)$ and $\frac{1}{9}$ .

$\log (64 x^{9}) - (\frac{1}{9} \log (4 x + 19))$

Step 1.5.2

Simplify $\frac{1}{9} \log (4 x + 19)$ by moving $\frac{1}{9}$ inside the logarithm.

$\log (64 x^{9}) - \log ({(4 x + 19)}^{\frac{1}{9}})$

Step 2

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

$\log (\frac{64 x^{9}}{{(4 x + 19)}^{\frac{1}{9}}})$