Write as a Single Logarithm log base b of x^2+1-1/2 log base b of x-2x^3

$\log_{b} (x^{2} + 1) - \frac{1}{2} \log_{b} (x - 2) x^{3}$

Multiply $- \frac{1}{2} (\log_{b} (x - 2) x^{3})$ .

Reorder $\log_{b} (x - 2)$ and $\frac{1}{2}$ .

$\log_{b} (x^{2} + 1) - (\frac{1}{2} \log_{b} (x - 2)) x^{3}$

Simplify $\frac{1}{2} \log_{b} (x - 2)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{b} (x^{2} + 1) - \log_{b} ({(x - 2)}^{\frac{1}{2}}) x^{3}$

Reorder factors in $\log_{b} (x^{2} + 1) - \log_{b} ({(x - 2)}^{\frac{1}{2}}) x^{3}$ .

$\log_{b} (x^{2} + 1) - x^{3} \log_{b} ({(x - 2)}^{\frac{1}{2}})$