<s>[INST] Simplify $\frac14\log(3x+1)+\frac12\log(x-3)-\frac12\log(x^4-x^2-10)$ $$\frac14\log(3x+1)+\frac12\log(x-3)-\frac12\log(x^4-x^2-10)$$
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.
 [/INST] 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:
$$\log\left(\dfrac{{{\sqrt[4]{3x+1}\sqrt{(x-3)(x^4-x^2-10)}}}}{x^4-x^2-10}\right)$$
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