Precalculus Examples

$4 \log_{3} (27) + \frac{1}{3} \log_{3} (x + 4) - 5 \log_{3} (x + 2) - \log_{3} (x)$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Logarithm base $3$ of $27$ is $3$ .

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

Step 1.2

Multiply $4$ by $3$ .

$12 + \frac{1}{3} \cdot \log_{3} (x + 4) - 5 \log_{3} (x + 2) - \log_{3} (x)$

Step 1.3

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

$12 + \log_{3} ({(x + 4)}^{\frac{1}{3}}) - 5 \log_{3} (x + 2) - \log_{3} (x)$

Step 1.4

Simplify $- 5 \log_{3} (x + 2)$ by moving $5$ inside the logarithm.

$12 + \log_{3} ({(x + 4)}^{\frac{1}{3}}) - \log_{3} ({(x + 2)}^{5}) - \log_{3} (x)$

Step 2

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

$12 + \log_{3} (\frac{{(x + 4)}^{\frac{1}{3}}}{{(x + 2)}^{5}}) - \log_{3} (x)$

Step 3

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

$12 + \log_{3} (\frac{\frac{{(x + 4)}^{\frac{1}{3}}}{{(x + 2)}^{5}}}{x})$

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$12 + \log_{3} (\frac{{(x + 4)}^{\frac{1}{3}}}{{(x + 2)}^{5}} \cdot \frac{1}{x})$

Step 4.2

Combine.

$12 + \log_{3} (\frac{{(x + 4)}^{\frac{1}{3}} \cdot 1}{{(x + 2)}^{5} x})$

Step 4.3

Multiply ${(x + 4)}^{\frac{1}{3}}$ by $1$ .

$12 + \log_{3} (\frac{{(x + 4)}^{\frac{1}{3}}}{{(x + 2)}^{5} x})$

Step 5

Reorder factors in $12 + \log_{3} (\frac{{(x + 4)}^{\frac{1}{3}}}{{(x + 2)}^{5} x})$ .

$12 + \log_{3} (\frac{{(x + 4)}^{\frac{1}{3}}}{x {(x + 2)}^{5}})$