Simplify $\frac14\log(3x+1)+\frac12\log(x-3)-\frac12\log(x^4-x^2-10)$

Question

$$\frac14\log(3x+1)+\frac12\log(x-3)-\frac12\log(x^4-x^2-10)$$ I was able to find $$\log(3x+1)^{1/4}(x-3)^{1/2}-\log(x^4-x^2-10)^{1/2}$$ through distributing the $^{1/2}$ but this answer is incorrect.

Answer 1

I do not know how you would get a "simplified answer", but:

$$\dfrac 14\log(3x+1)+\dfrac 12 \log(x-3)-\dfrac 12\log(x^4-x^2-10)$$

Using the property $\log a^b=b\log a$:

$$\log{\sqrt[4]{3x+1}}+ \log{\sqrt{x-3}}-\log\sqrt{{x^4-x^2-10}}$$

Using the property $\log a+\log b=\log ab$:

$$\log{\sqrt[4]{3x+1}}+ \log{\sqrt{x-3}}-\log\sqrt{{x^4-x^2-10}}=\log\left(\dfrac{{{\sqrt[4]{3x+1}\sqrt{x-3}}}}{\sqrt{x^4-x^2-10}}\right)$$

Rationalizing:

$$\log\left(\dfrac{{{\sqrt[4]{3x+1}\sqrt{x-3}}}}{\sqrt{x^4-x^2-10}}\cdot \dfrac {\sqrt{x^4-x^2-10}}{\sqrt{x^4-x^2-10}}\right)=\log\left(\dfrac{{{\sqrt[4]{3x+1}\sqrt{(x-3)(x^4-x^2-10)}}}}{x^4-x^2-10}\right)$$

The answer does not seem to "simplify" further than the following:

$$\log\left(\dfrac{{{\sqrt[4]{3x+1}\sqrt{(x-3)(x^4-x^2-10)}}}}{x^4-x^2-10}\right)$$

Answer 2

We can take

$$\frac14\log(3x+1)+\frac12\log(x-3)-\frac12\log(x^4-x^2-10)=\frac14\log(3x+1)+\frac14\log(x-3)^2-\frac14\log(x^4-x^2-10)^2=\frac14\log \left[\frac{(3x+1)(x-3)^2}{(x^4-x^2-10)^2}\right]$$

but it doesn’t seem possible to simplify further.