Expand the Logarithmic Expression natural log of (5x square root of 2-7x)/(3(x-4)^8)

$\ln (\frac{5 x \sqrt{2 - 7 x}}{3 {(x - 4)}^{8}})$

Rewrite $\ln (\frac{5 x \sqrt{2 - 7 x}}{3 {(x - 4)}^{8}})$ as $\ln (5 x \sqrt{2 - 7 x}) - \ln (3 {(x - 4)}^{8})$ .

$\ln (5 x \sqrt{2 - 7 x}) - \ln (3 {(x - 4)}^{8})$

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

$\ln (5 x {(2 - 7 x)}^{\frac{1}{2}}) - \ln (3 {(x - 4)}^{8})$

Rewrite $\ln (5 x {(2 - 7 x)}^{\frac{1}{2}})$ as $\ln (5 x) + \ln ({(2 - 7 x)}^{\frac{1}{2}})$ .

$\ln (5 x) + \ln ({(2 - 7 x)}^{\frac{1}{2}}) - \ln (3 {(x - 4)}^{8})$

Rewrite $\ln (5 x)$ as $\ln (5) + \ln (x)$ .

$\ln (5) + \ln (x) + \ln ({(2 - 7 x)}^{\frac{1}{2}}) - \ln (3 {(x - 4)}^{8})$

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

$\ln (5) + \ln (x) + \frac{1}{2} \ln (2 - 7 x) - \ln (3 {(x - 4)}^{8})$

Rewrite $\ln (3 {(x - 4)}^{8})$ as $\ln (3) + \ln ({(x - 4)}^{8})$ .

$\ln (5) + \ln (x) + \frac{1}{2} \ln (2 - 7 x) - (\ln (3) + \ln ({(x - 4)}^{8}))$

Expand $\ln ({(x - 4)}^{8})$ by moving $8$ outside the logarithm.

$\ln (5) + \ln (x) + \frac{1}{2} \ln (2 - 7 x) - (\ln (3) + 8 \ln (x - 4))$

Simplify each term.

Combine $\frac{1}{2}$ and $\ln (2 - 7 x)$ .

$\ln (5) + \ln (x) + \frac{\ln (2 - 7 x)}{2} - (\ln (3) + 8 \ln (x - 4))$

Apply the distributive property.

$\ln (5) + \ln (x) + \frac{\ln (2 - 7 x)}{2} - \ln (3) - (8 \ln (x - 4))$

Multiply $8$ by $- 1$ .

$\ln (5) + \ln (x) + \frac{\ln (2 - 7 x)}{2} - \ln (3) - 8 \ln (x - 4)$

To write $\ln (5)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\ln (5) \cdot \frac{2}{2} + \frac{\ln (2 - 7 x)}{2} + \ln (x) - \ln (3) - 8 \ln (x - 4)$

Combine $\ln (5)$ and $\frac{2}{2}$ .

$\frac{\ln (5) \cdot 2}{2} + \frac{\ln (2 - 7 x)}{2} + \ln (x) - \ln (3) - 8 \ln (x - 4)$

Combine the numerators over the common denominator.

$\frac{\ln (5) \cdot 2 + \ln (2 - 7 x)}{2} + \ln (x) - \ln (3) - 8 \ln (x - 4)$

Move $2$ to the left of $\ln (5)$ .

$\frac{2 \ln (5) + \ln (2 - 7 x)}{2} + \ln (x) - \ln (3) - 8 \ln (x - 4)$

To write $- 8 \ln (x - 4)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2 \ln (5) + \ln (2 - 7 x)}{2} - 8 \ln (x - 4) \cdot \frac{2}{2} + \ln (x) - \ln (3)$

Combine $- 8 \ln (x - 4)$ and $\frac{2}{2}$ .

$\frac{2 \ln (5) + \ln (2 - 7 x)}{2} + \frac{- 8 \ln (x - 4) \cdot 2}{2} + \ln (x) - \ln (3)$

Combine the numerators over the common denominator.

$\frac{2 \ln (5) + \ln (2 - 7 x) - 8 \ln (x - 4) \cdot 2}{2} + \ln (x) - \ln (3)$

Multiply $2$ by $- 8$ .

$\frac{2 \ln (5) + \ln (2 - 7 x) - 16 \ln (x - 4)}{2} + \ln (x) - \ln (3)$