Algebra

$\large a^{2x} = b^{x + 1}$

$\large \implies a^{2x} = b^x . b$

$\large \implies \frac{a^{2x}}{b^x} = b$

$\large \implies \left[\frac{a^2}{b} \right]^x = b$

By taking natural logarithm

$\large \implies x \left[log(a^2) - log(b) \right] = log b$

$\large \therefore x = \frac{log b}{log(a^2) - logb}$

$\large \implies x = \frac{log b}{2 loga - logb}$