Precalculus Examples

$\ln (\sqrt{x^{3} y^{2} z e^{\frac{3}{2}}})$

Step 1

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

$\ln ({(x^{3} y^{2} z e^{\frac{3}{2}})}^{\frac{1}{2}})$

Step 2

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

$\frac{1}{2} \ln (x^{3} y^{2} z e^{\frac{3}{2}})$

Step 3

Rewrite $\ln (x^{3} y^{2} z e^{\frac{3}{2}})$ as $\ln (x^{3} y^{2} z) + \ln (e^{\frac{3}{2}})$ .

$\frac{1}{2} (\ln (x^{3} y^{2} z) + \ln (e^{\frac{3}{2}}))$

Step 4

Rewrite $\ln (x^{3} y^{2} z)$ as $\ln (x^{3} y^{2}) + \ln (z)$ .

$\frac{1}{2} (\ln (x^{3} y^{2}) + \ln (z) + \ln (e^{\frac{3}{2}}))$

Step 5

Rewrite $\ln (x^{3} y^{2})$ as $\ln (x^{3}) + \ln (y^{2})$ .

$\frac{1}{2} (\ln (x^{3}) + \ln (y^{2}) + \ln (z) + \ln (e^{\frac{3}{2}}))$

Step 6

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

$\frac{1}{2} (3 \ln (x) + \ln (y^{2}) + \ln (z) + \ln (e^{\frac{3}{2}}))$

Step 7

Expand $\ln (y^{2})$ by moving $2$ outside the logarithm.

$\frac{1}{2} (3 \ln (x) + 2 \ln (y) + \ln (z) + \ln (e^{\frac{3}{2}}))$

Step 8

Expand $\ln (e^{\frac{3}{2}})$ by moving $\frac{3}{2}$ outside the logarithm.

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

Step 9

The natural logarithm of $e$ is $1$ .

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

Step 10

Multiply $\frac{3}{2}$ by $1$ .

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

Step 11

Apply the distributive property.

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

Step 12

Simplify.

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Step 12.1

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

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Step 12.1.1

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

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

Step 12.1.2

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

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

Step 12.2

Cancel the common factor of $2$ .

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Step 12.2.1

Factor $2$ out of $2 \ln (y)$ .

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

Step 12.2.2

Cancel the common factor.

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

Step 12.2.3

Rewrite the expression.

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

Step 12.3

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

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

Step 12.4

Multiply $\frac{1}{2} \cdot \frac{3}{2}$ .

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Step 12.4.1

Multiply $\frac{1}{2}$ by $\frac{3}{2}$ .

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

Step 12.4.2

Multiply $2$ by $2$ .

$\frac{3 \ln (x)}{2} + \ln (y) + \frac{\ln (z)}{2} + \frac{3}{4}$