What is the sum of all possible values of x such that $$8^x\ =\ 3^{x^2}$$?

The equation given is,

$$8^x\ =\ 3^{x^2}$$

Applying log on both sides, we get,

$$\log8^x\ =\ \log3^{x^2}$$

$$x\log8\ =\ x^2\log3$$

$$\ x^2\log3\ -\ x\log8\ =\ 0$$

$$\ x\left(x\log3\ -\ \log8\right)\ =\ 0$$

So, the solutions of the above equation are x = 0 and xlog3 = log8, which can also be written as $$\ x\ =\ \dfrac{\log8}{\log3}\ =\ \log_38$$

The sum of all possible values is $$0\ +\ \log_38\ =\ \log_38$$ which can also be written as $$\log_32\ +\ \log_34$$.

The correct answer is option B.