Precalculus Examples

$\ln (\sqrt{6 x^{7} y^{3}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 x^{7} y^{3}}$ as ${(6 x^{7} y^{3})}^{\frac{1}{2}}$ .

$\ln ({(6 x^{7} y^{3})}^{\frac{1}{2}})$

Step 2

Expand $\ln ({(6 x^{7} y^{3})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \ln (6 x^{7} y^{3})$

Step 3

Rewrite $\ln (6 x^{7} y^{3})$ as $\ln (6 x^{7}) + \ln (y^{3})$ .

$\frac{1}{2} (\ln (6 x^{7}) + \ln (y^{3}))$

Step 4

Rewrite $\ln (6 x^{7})$ as $\ln (6) + \ln (x^{7})$ .

$\frac{1}{2} (\ln (6) + \ln (x^{7}) + \ln (y^{3}))$

Step 5

Expand $\ln (x^{7})$ by moving $7$ outside the logarithm.

$\frac{1}{2} (\ln (6) + 7 \ln (x) + \ln (y^{3}))$

Step 6

Expand $\ln (y^{3})$ by moving $3$ outside the logarithm.

$\frac{1}{2} (\ln (6) + 7 \ln (x) + 3 \ln (y))$

Step 7

Simplify each term.

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Step 7.1

Rewrite $\ln (6)$ as $\ln (2 (3))$ .

$\frac{1}{2} (\ln (2 (3)) + 7 \ln (x) + 3 \ln (y))$

Step 7.2

Rewrite $\ln (2 (3))$ as $\ln (2) + \ln (3)$ .

$\frac{1}{2} (\ln (2) + \ln (3) + 7 \ln (x) + 3 \ln (y))$

Step 8

Apply the distributive property.

$\frac{1}{2} \ln (2) + \frac{1}{2} \ln (3) + \frac{1}{2} (7 \ln (x)) + \frac{1}{2} (3 \ln (y))$

Step 9

Simplify.

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Step 9.1

Combine $\frac{1}{2}$ and $\ln (2)$ .

$\frac{\ln (2)}{2} + \frac{1}{2} \ln (3) + \frac{1}{2} (7 \ln (x)) + \frac{1}{2} (3 \ln (y))$

Step 9.2

Combine $\frac{1}{2}$ and $\ln (3)$ .

$\frac{\ln (2)}{2} + \frac{\ln (3)}{2} + \frac{1}{2} (7 \ln (x)) + \frac{1}{2} (3 \ln (y))$

Step 9.3

Multiply $\frac{1}{2} (7 \ln (x))$ .

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Step 9.3.1

Combine $7$ and $\frac{1}{2}$ .

$\frac{\ln (2)}{2} + \frac{\ln (3)}{2} + \frac{7}{2} \ln (x) + \frac{1}{2} (3 \ln (y))$

Step 9.3.2

Combine $\frac{7}{2}$ and $\ln (x)$ .

$\frac{\ln (2)}{2} + \frac{\ln (3)}{2} + \frac{7 \ln (x)}{2} + \frac{1}{2} (3 \ln (y))$

Step 9.4

Multiply $\frac{1}{2} (3 \ln (y))$ .

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Step 9.4.1

Combine $3$ and $\frac{1}{2}$ .

$\frac{\ln (2)}{2} + \frac{\ln (3)}{2} + \frac{7 \ln (x)}{2} + \frac{3}{2} \ln (y)$

Step 9.4.2

Combine $\frac{3}{2}$ and $\ln (y)$ .

$\frac{\ln (2)}{2} + \frac{\ln (3)}{2} + \frac{7 \ln (x)}{2} + \frac{3 \ln (y)}{2}$