A logarithm equation

Algebra Level 2

The value of x\displaystyle{x} that satisfies the equation log2(2x1+3x+1)=2xlog2(3x)\log_{2}(2^{x-1} + 3^{x+1}) = 2x - \log_{2}(3^x) can be expressed as logab\displaystyle{\log_{a}b}. What is the value of a+b\displaystyle{a+b}?

\quad Note: aa and bb are both fractions and their sum is an integer.

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