Question 691401
{{{ 2log(9,x) = 1/2 + log(9,(5x + 18)) }}}
{{{ 2log(9,x) - log(9,(5x + 18)) = 1/2 }}}
{{{ log(9,x^2) - log(9,(5x + 18)) = 1/2 }}}    (using {{{ alog(b)=log(b^a) }}})
{{{ log(9,x^2 / (5x + 18)) = 1/2 }}}    (using {{{ log(a)-log(b)=log(a/b) }}})
{{{ x^2 / (5x + 18) = 9^(1/2) }}}    (taking anti-logarithms)
{{{ x^2 / (5x + 18) = sqrt(9) }}}    (definition of fractional powers)
{{{ x^2 / (5x + 18) = 3 }}}
{{{ x^2 = 3(5x + 18) }}}
{{{ x^2 = 15x + 54 }}}
{{{ x^2 - 15x - 54 = 0 }}} (which is a quadratic in x ...)
{{{ (x-18)(x+3) = 0 }}} (... and it factorises very nicely)
{{{ x=-3 }}} or {{{ x=18 }}}

Now original question involved {{{ 2log(9,x) }}} and you cannot take the logarithm of a negative number, so we can assume that {{{ x > 0 }}} and eliminate any negative solutions.

So, answer is {{{ x=18 }}}