Precalculus Examples

$\ln (e^{\frac{2}{5}} x^{7} y^{\frac{3}{2}})$

Step 1

Rewrite $\ln (e^{\frac{2}{5}} x^{7} y^{\frac{3}{2}})$ as $\ln (e^{\frac{2}{5}} x^{7}) + \ln (y^{\frac{3}{2}})$ .

$\ln (e^{\frac{2}{5}} x^{7}) + \ln (y^{\frac{3}{2}})$

Step 2

Rewrite $\ln (e^{\frac{2}{5}} x^{7})$ as $\ln (e^{\frac{2}{5}}) + \ln (x^{7})$ .

$\ln (e^{\frac{2}{5}}) + \ln (x^{7}) + \ln (y^{\frac{3}{2}})$

Step 3

Expand $\ln (e^{\frac{2}{5}})$ by moving $\frac{2}{5}$ outside the logarithm.

$\frac{2}{5} \ln (e) + \ln (x^{7}) + \ln (y^{\frac{3}{2}})$

Step 4

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

$\frac{2}{5} \ln (e) + 7 \ln (x) + \ln (y^{\frac{3}{2}})$

Step 5

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

$\frac{2}{5} \ln (e) + 7 \ln (x) + \frac{3}{2} \ln (y)$

Step 6

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

$\frac{2}{5} \cdot 1 + 7 \ln (x) + \frac{3}{2} \ln (y)$

Step 7

Multiply $\frac{2}{5}$ by $1$ .

$\frac{2}{5} + 7 \ln (x) + \frac{3}{2} \ln (y)$

Step 8

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

$\frac{2}{5} + 7 \ln (x) + \frac{3 \ln (y)}{2}$