We can write -1/2 logq(81q^2 )=p as logq[(81q^2 )^-1/2], based on the laws of logs. This can be simplified to logq(1/9q) = P. This can be further simplified to logq(1/9) + logq(1/q). We know that logq(1/q) is simply -1, so we can rewrite the equation as logq(1/9) - 1 = P.Bringing 1 over to the other side gives logq(1/9) = P + 1. We can use the change of base formula to write logq(1/9) as log(1/9)/logq, giving the equation log(1/9)/logq = P + 1. Now we multiply both sides by logq to clear the equation of the denominator, resulting in log(1/9) = (P + 1)logq. We divide by (P + 1) in order to isolate q, giving logq = log(1/9)/(P + 1). Now we simply write the equation in exponential form, and find that 10^[log(1/9)/(P + 1)] = q.
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