A logarithm equation

Algebra Level 2

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

\(\quad\) Note: \(a\) and \(b\) are both fractions and their sum is an integer.

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