OCR C2 2015 Question 8: (a) Use logarithms to solve the equation 2^(n-3) = 18,000 , giving your answer correct to 3 significant figures. (b) Solve the simultaneous equations log2(x) + log2(y) = 8 & log2(x^2/y) = 7.
(a)This question actually tells us what to do. It is very hard to miss "use logarithms to solve...". So our first step is going to be to use logs (especially as we can also see that is very little else that we can do and we know that we are going to need to use logs at some point).Using base 2 (for reasons which will become obvious soon), we get the following:log2(2^n-3) = log2(18000) Now using some basic log rules... (n-3)log2(2) = log2(18000) Now hopefully the use of base 2 makes sense as log22 = 1 So... n - 3 = log218000 which gives n = 3 + log218000 However, this is not our final answer. When we read the question again we see that it wants us to give our answer to 3 significant figures. So using the fact that log218000 is 14.1357... we get our final answer of n = 17.1 (3sf). (b)This part of the question is definitely more challenging, but is still very doable so long as you remember your basic log rules and methods for solving simultaneous equations. Using log rules on the second equation we get 2log2x - log2y = 7. Now we can easily see that we can eliminate the y terms by adding our 2 equations together giving 3log2x = 15, and hence that x = 25 = 32 using inverse logs. From here it is a substitution into either equation get x = 32, y = 8.
