Question 970559
i would solve this as follows:


start with 2^(a+3) = 3^(2a-1)


take the log of both sides of the equation to get:


log(2^(a+3)) = log(3^(2a-1)


since log(x^a) = a*log(x), your equation of log(2^(a+3)) = log(3^(2a-1)
becomes:


(a+3)*log(2) = (2a-1)*log(3)


divide both sides of this equation by log(2) to get:


a+3 = (2a-1)*log(3)/log(2)


to simplify how this looks, we will temporarily let b = log(3)/log(2).


the equation of a+3 = (2a-1)*log(3)/log(2) becomes a+3 = (2a-1)*b


perform the multiplication on the right side of the equation to get:

 
a+3 = 2ab-b


subtract a from both sides of this equation and add b to both sides of this eqaution to get:


b+3 = 2ab-a


factor out the a on the right side of this equation to get:


b+3 = (2b-1)*a


divide both sides of this eqaution by (2b-1) to get:


(b+3)/(2b-1) = a


we had previously let b = log(3)/log(2).


now we replace b with log(3)/log(2) to get:


(b+3)/(2b-1) = a becomes:


(log(3)/log(2)+3)/(2*log(3)/log(2)-1) = a


use your calculator to solve for a to get a = 2.112958972.


that's your solution.


replace a in the original equation with 2.112958972 and you get:


log(2^(a+3)) = log(3^(2a-1) becomes:


log(2^(2.112958972+3) = log(3^(2*2.112958972-1) which becomes:


34.60620827 = 34.60620827 which is true.


this confirms the solution is good.


the solution is a = 2.112958972