Precalculus Examples

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

Step 1

Multiply $\frac{1}{5} (\log_{3} (x + 2) \log_{3} (z - \log_{3} (y)))$ .

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Step 1.1

Reorder $\log_{3} (x + 2)$ and $\frac{1}{5}$ .

$\frac{1}{5} \log_{3} (x + 2) \log_{3} (z - \log_{3} (y))$

Step 1.2

Simplify $\frac{1}{5} \log_{3} (x + 2)$ by moving $\frac{1}{5}$ inside the logarithm.

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

Step 2

Rewrite $\log_{3} ({(x + 2)}^{\frac{1}{5}})$ using the change of base formula.

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Step 2.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{3} (z - \log_{3} (y))$

Step 2.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log ({(x + 2)}^{\frac{1}{5}})}{\log (3)} \log_{3} (z - \log_{3} (y))$

Step 3

Rewrite $\log_{3} (z - \log_{3} (y))$ using the change of base formula.

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Step 3.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{\log ({(x + 2)}^{\frac{1}{5}})}{\log (3)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 3.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log ({(x + 2)}^{\frac{1}{5}})}{\log (3)} \cdot \frac{\log (z - \log_{3} (y))}{\log (3)}$

Step 4

Combine.

$\frac{\log ({(x + 2)}^{\frac{1}{5}}) \log (z - \log_{3} (y))}{\log (3) \log (3)}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $\log_{3} (y)$ using the change of base formula.

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Step 5.1.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{\log ({(x + 2)}^{\frac{1}{5}}) \log (z - \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)})}{\log (3) \log (3)}$

Step 5.1.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log ({(x + 2)}^{\frac{1}{5}}) \log (z - \frac{\log (y)}{\log (3)})}{\log (3) \log (3)}$

Step 5.2

To write $z$ as a fraction with a common denominator, multiply by $\frac{\log (3)}{\log (3)}$ .

$\frac{\log ({(x + 2)}^{\frac{1}{5}}) \log (\frac{z \log (3)}{\log (3)} - \frac{\log (y)}{\log (3)})}{\log (3) \log (3)}$

Step 5.3

Combine the numerators over the common denominator.

$\frac{\log ({(x + 2)}^{\frac{1}{5}}) \log (\frac{z \log (3) - \log (y)}{\log (3)})}{\log (3) \log (3)}$

Step 6

Simplify the denominator.

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Step 6.1

Raise $\log (3)$ to the power of $1$ .

$\frac{\log ({(x + 2)}^{\frac{1}{5}}) \log (\frac{z \log (3) - \log (y)}{\log (3)})}{\log^{1} (3) \log (3)}$

Step 6.2

Raise $\log (3)$ to the power of $1$ .

$\frac{\log ({(x + 2)}^{\frac{1}{5}}) \log (\frac{z \log (3) - \log (y)}{\log (3)})}{\log^{1} (3) \log^{1} (3)}$

Step 6.3

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

$\frac{\log ({(x + 2)}^{\frac{1}{5}}) \log (\frac{z \log (3) - \log (y)}{\log (3)})}{{\log (3)}^{1 + 1}}$

Step 6.4

Add $1$ and $1$ .

$\frac{\log ({(x + 2)}^{\frac{1}{5}}) \log (\frac{z \log (3) - \log (y)}{\log (3)})}{\log^{2} (3)}$