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# Heinemann C2 Book Question watch

1. Hi everyone,

Sad first post I know, but: my teacher and I both can't answer this question and I thouht some great TSRer could have a go, so here goes:

HMM: C2: page 47: question 10:
Prove that if a^x = b^y = (ab)^xy then x + y = 1
Any help on that would be really appreciated.

Thanks,
Jimbo
2. Welcome to the forum.

a^x = b^y = (ab)^xy

Let's break it down into:
a^x=b^y (1)
a^x=(ab)^(xy) (2)
b^y=(ab)^(xy) (3)

xlna=ylnb so y=(xlna)/(lnb) or x=(ylnb)/(lna) (from 1)

xlna=xyln(ab)=x[(xlna)/(lnb)][lnab] (from 2)
xlna=(x^2)(lna)(lnab)/(lnb)
1=xln(ab)/lnb
x=(lnb)/(lnab)
x=(lnb)/(lna+lnb)

ylnb=xyln(ab) (from 3)
ylnb=(y^2)(lnb/lna).(lnab)
lna/lnab=y(lnab)
y=(lna)/(lna+lnb)

x+y=(lna)/(lna+lnb)+(lnb)/(lna+lnb) (from our 2 bolded results)
x+y=(lna+lnb)/(lna+lnb)
x+y=1
3. WOW, thanks for being so quick, that really was a great welcome. Only thing, does your use of "ln" as opposed to "log" make any difference, as we've not learnt about logs yet?

But thanks very much!

Jimbo
4. (Original post by jimbo)
WOW, thanks for being so quick, that really was a great welcome. Only thing, does your use of "ln" as opposed to "log" make any difference, as we've not learnt about logs yet?

But thanks very much!

Jimbo
I've edited my post to make it a little more readable, it was a bit random at first
'ln' just means 'log to the base e' or 'natural logarithm', it doesn't really make any difference which base you use though in this case as they cancel out to give x+y=1 and we don't actually numerically evaluate any of the logarithms.
Are logs in C2? I'm just finishing doing the old syllabus so i've just done it in the easiest way I can think of.
5. Prove that if a^x = b^y = (ab)^xy then x + y = 1
(ab)xy
= axybxy
= (ax)y x (by)x
= (by)y x (by)x (since ax = by)
= by2 x byx
= b(y^2 + yx) = by

Comparing powers
y^2 + yx = y
Divide by y giving; y + x = 1

Wow, i've improved in maths. When I first looked at that a few months ago my jaw literally dropped.
6. Great, that's just perfect. Thought it might be multibased as it were, and "ln" looks very academic.

Thanks a lot matey.

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