Precalculus Examples

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

Step 1

Rewrite $\ln (\sqrt{x + 1} \cdot {(y^{2} + 1)}^{\frac{3}{2}} \cdot e^{\frac{3}{2}})$ as $\ln (\sqrt{x + 1} \cdot {(y^{2} + 1)}^{\frac{3}{2}}) + \ln (e^{\frac{3}{2}})$ .

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

Step 2

Rewrite $\ln (\sqrt{x + 1} \cdot {(y^{2} + 1)}^{\frac{3}{2}})$ as $\ln (\sqrt{x + 1}) + \ln ({(y^{2} + 1)}^{\frac{3}{2}})$ .

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

Step 3

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

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

Step 4

Expand $\ln ({(x + 1)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify each term.

Tap for more steps...

Step 9.1

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

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

Step 9.2

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

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

Step 10

Combine the numerators over the common denominator.

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