Expand the Logarithmic Expression natural log of (x^3)/( square root of x^2-25)

$\ln (\frac{x^{3}}{\sqrt{x^{2} - 25}})$

Rewrite $\ln (\frac{x^{3}}{\sqrt{x^{2} - 25}})$ as $\ln (x^{3}) - \ln (\sqrt{x^{2} - 25})$ .

$\ln (x^{3}) - \ln (\sqrt{x^{2} - 25})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} - 25}$ as ${(x^{2} - 25)}^{\frac{1}{2}}$ .

$\ln (x^{3}) - \ln ({(x^{2} - 25)}^{\frac{1}{2}})$

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

$3 \ln (x) - \ln ({(x^{2} - 25)}^{\frac{1}{2}})$

Expand $\ln ({(x^{2} - 25)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$3 \ln (x) - (\frac{1}{2} \ln (x^{2} - 25))$

Combine $\frac{1}{2}$ and $\ln (x^{2} - 25)$ .

$3 \ln (x) - \frac{\ln (x^{2} - 25)}{2}$

To write $3 \ln (x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 \ln (x) \cdot \frac{2}{2} - \frac{\ln (x^{2} - 25)}{2}$

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

$\frac{3 \ln (x) \cdot 2}{2} - \frac{\ln (x^{2} - 25)}{2}$

Combine the numerators over the common denominator.

$\frac{3 \ln (x) \cdot 2 - \ln (x^{2} - 25)}{2}$

Multiply $2$ by $3$ .

$\frac{6 \ln (x) - \ln (x^{2} - 25)}{2}$