Precalculus Examples

$\ln (\frac{4 d f^{5}}{\sqrt[8]{1 - 3 d}})$

Step 1

Rewrite $\ln (\frac{4 d f^{5}}{\sqrt[8]{1 - 3 d}})$ as $\ln (4 d f^{5}) - \ln (\sqrt[8]{1 - 3 d})$ .

$\ln (4 d f^{5}) - \ln (\sqrt[8]{1 - 3 d})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[8]{1 - 3 d}$ as ${(1 - 3 d)}^{\frac{1}{8}}$ .

$\ln (4 d f^{5}) - \ln ({(1 - 3 d)}^{\frac{1}{8}})$

Step 3

Expand $\ln ({(1 - 3 d)}^{\frac{1}{8}})$ by moving $\frac{1}{8}$ outside the logarithm.

$\ln (4 d f^{5}) - (\frac{1}{8} \ln (1 - 3 d))$

Step 4

Combine $\frac{1}{8}$ and $\ln (1 - 3 d)$ .

$\ln (4 d f^{5}) - \frac{\ln (1 - 3 d)}{8}$

Step 5

Rewrite $\ln (4 d f^{5})$ as $\ln (4 d) + \ln (f^{5})$ .

$\ln (4 d) + \ln (f^{5}) - \frac{\ln (1 - 3 d)}{8}$

Step 6

Rewrite $\ln (4 d)$ as $\ln (4) + \ln (d)$ .

$\ln (4) + \ln (d) + \ln (f^{5}) - \frac{\ln (1 - 3 d)}{8}$

Step 7

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

$\ln (4) + \ln (d) + 5 \ln (f) - \frac{\ln (1 - 3 d)}{8}$

Step 8

Simplify each term.

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Step 8.1

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

$\ln (2^{2}) + \ln (d) + 5 \ln (f) - \frac{\ln (1 - 3 d)}{8}$

Step 8.2

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

$2 \ln (2) + \ln (d) + 5 \ln (f) - \frac{\ln (1 - 3 d)}{8}$

Step 9

To write $2 \ln (2)$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\ln (d) + 5 \ln (f) + 2 \ln (2) \cdot \frac{8}{8} - \frac{\ln (1 - 3 d)}{8}$

Step 10

Combine $2 \ln (2)$ and $\frac{8}{8}$ .

$\ln (d) + 5 \ln (f) + \frac{2 \ln (2) \cdot 8}{8} - \frac{\ln (1 - 3 d)}{8}$

Step 11

Combine the numerators over the common denominator.

$\ln (d) + 5 \ln (f) + \frac{2 \ln (2) \cdot 8 - \ln (1 - 3 d)}{8}$

Step 12

Multiply $8$ by $2$ .

$\ln (d) + 5 \ln (f) + \frac{16 \ln (2) - \ln (1 - 3 d)}{8}$