solve for x, in the form x = loga/logb for 2^(4x - 1) = 3^(5-2x) (taken from OCR June 2014 C2)

We can take logs of both sides straight away, and using the log rule that alogb = log(b^a)So(4x-1) log(2) = (5 - 2x) log(3)We can expand the bracketsso (4x)log(2) - log(2) = 5log(3) - 2xlog(3)We can group the termsso x(4log(2) + 2log(3)) = 5log(3) + log(2)Hence x = (5log(3) + log(2))/(4log(2) + 2log(3))Now re-read the question. We have not finished: this isn't in the desired form. We must have single logs on top and bottom.The numerator can be changed to log(3^5) + log(2) Using the log rule that alogb = log(b^a) (which we used at the start)Now we can use the rule that log(y) + log(z) = log(yz) when the bases are the same. The bases are the same here (base 10).So the numerator becomes log(3^(5)2) = log(486)By following the same process for the denominator, we find log(2^(4)3^(2)) = log(144)These can't simplify as they contain factors with different values (e.g. log(16) can be simplified to 4log(2) as 16 = 2222 and is not a mixture of numbers unlike e.g. 18 = 23*3)Hence the final answer is x = log(486)/log(144)