Precalculus Examples

$3 \log (\frac{9}{\sqrt{x + 3}})$

Step 1

Multiply $\frac{9}{\sqrt{x + 3}}$ by $\frac{\sqrt{x + 3}}{\sqrt{x + 3}}$ .

$3 \log (\frac{9}{\sqrt{x + 3}} \cdot \frac{\sqrt{x + 3}}{\sqrt{x + 3}})$

Step 2

Combine and simplify the denominator.

Tap for more steps...

Step 2.1

Multiply $\frac{9}{\sqrt{x + 3}}$ by $\frac{\sqrt{x + 3}}{\sqrt{x + 3}}$ .

$3 \log (\frac{9 \sqrt{x + 3}}{\sqrt{x + 3} \sqrt{x + 3}})$

Step 2.2

Raise $\sqrt{x + 3}$ to the power of $1$ .

$3 \log (\frac{9 \sqrt{x + 3}}{{\sqrt{x + 3}}^{1} \sqrt{x + 3}})$

Step 2.3

Raise $\sqrt{x + 3}$ to the power of $1$ .

$3 \log (\frac{9 \sqrt{x + 3}}{{\sqrt{x + 3}}^{1} {\sqrt{x + 3}}^{1}})$

Step 2.4

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

$3 \log (\frac{9 \sqrt{x + 3}}{{\sqrt{x + 3}}^{1 + 1}})$

Step 2.5

Add $1$ and $1$ .

$3 \log (\frac{9 \sqrt{x + 3}}{{\sqrt{x + 3}}^{2}})$

Step 2.6

Rewrite ${\sqrt{x + 3}}^{2}$ as $x + 3$ .

Tap for more steps...

Step 2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 3}$ as ${(x + 3)}^{\frac{1}{2}}$ .

$3 \log (\frac{9 \sqrt{x + 3}}{{({(x + 3)}^{\frac{1}{2}})}^{2}})$

Step 2.6.2

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

$3 \log (\frac{9 \sqrt{x + 3}}{{(x + 3)}^{\frac{1}{2} \cdot 2}})$

Step 2.6.3

Combine $\frac{1}{2}$ and $2$ .

$3 \log (\frac{9 \sqrt{x + 3}}{{(x + 3)}^{\frac{2}{2}}})$

Step 2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.6.4.1

Cancel the common factor.

$3 \log (\frac{9 \sqrt{x + 3}}{{(x + 3)}^{\frac{2}{2}}})$

Step 2.6.4.2

Rewrite the expression.

$3 \log (\frac{9 \sqrt{x + 3}}{{(x + 3)}^{1}})$

Step 2.6.5

Simplify.

$3 \log (\frac{9 \sqrt{x + 3}}{x + 3})$

Step 3

Simplify $3 \log (\frac{9 \sqrt{x + 3}}{x + 3})$ by moving $3$ inside the logarithm.

$\log ({(\frac{9 \sqrt{x + 3}}{x + 3})}^{3})$

Step 4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 4.1

Apply the product rule to $\frac{9 \sqrt{x + 3}}{x + 3}$ .

$\log (\frac{{(9 \sqrt{x + 3})}^{3}}{{(x + 3)}^{3}})$

Step 4.2

Apply the product rule to $9 \sqrt{x + 3}$ .

$\log (\frac{9^{3} {\sqrt{x + 3}}^{3}}{{(x + 3)}^{3}})$

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

Raise $9$ to the power of $3$ .

$\log (\frac{729 {\sqrt{x + 3}}^{3}}{{(x + 3)}^{3}})$

Step 5.2

Rewrite ${\sqrt{x + 3}}^{3}$ as $\sqrt{{(x + 3)}^{3}}$ .

$\log (\frac{729 \sqrt{{(x + 3)}^{3}}}{{(x + 3)}^{3}})$

Step 5.3

Factor out ${(x + 3)}^{2}$ .

$\log (\frac{729 \sqrt{{(x + 3)}^{2} (x + 3)}}{{(x + 3)}^{3}})$

Step 5.4

Pull terms out from under the radical.

$\log (\frac{729 ((x + 3) \sqrt{x + 3})}{{(x + 3)}^{3}})$

Step 5.5

Apply the distributive property.

$\log (\frac{729 (x \sqrt{x + 3} + 3 \sqrt{x + 3})}{{(x + 3)}^{3}})$

Step 5.6

Factor $\sqrt{x + 3}$ out of $x \sqrt{x + 3} + 3 \sqrt{x + 3}$ .

Tap for more steps...

Step 5.6.1

Factor $\sqrt{x + 3}$ out of $x \sqrt{x + 3}$ .

$\log (\frac{729 (\sqrt{x + 3} x + 3 \sqrt{x + 3})}{{(x + 3)}^{3}})$

Step 5.6.2

Factor $\sqrt{x + 3}$ out of $3 \sqrt{x + 3}$ .

$\log (\frac{729 (\sqrt{x + 3} x + \sqrt{x + 3} \cdot 3)}{{(x + 3)}^{3}})$

Step 5.6.3

Factor $\sqrt{x + 3}$ out of $\sqrt{x + 3} x + \sqrt{x + 3} \cdot 3$ .

$\log (\frac{729 (\sqrt{x + 3} (x + 3))}{{(x + 3)}^{3}})$

$\log (\frac{729 \sqrt{x + 3} (x + 3)}{{(x + 3)}^{3}})$

Step 6

Cancel the common factor of $x + 3$ and ${(x + 3)}^{3}$ .

Tap for more steps...

Step 6.1

Factor $x + 3$ out of $729 \sqrt{x + 3} (x + 3)$ .

$\log (\frac{(x + 3) (729 \sqrt{x + 3})}{{(x + 3)}^{3}})$

Step 6.2

Cancel the common factors.

Tap for more steps...

Step 6.2.1

Factor $x + 3$ out of ${(x + 3)}^{3}$ .

$\log (\frac{(x + 3) (729 \sqrt{x + 3})}{(x + 3) {(x + 3)}^{2}})$

Step 6.2.2

Cancel the common factor.

$\log (\frac{(x + 3) (729 \sqrt{x + 3})}{(x + 3) {(x + 3)}^{2}})$

Step 6.2.3

Rewrite the expression.

$\log (\frac{729 \sqrt{x + 3}}{{(x + 3)}^{2}})$