Express log value from other log values

Sep 2016
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0
Muskoka
q5afch6.pngI got a really messy answer (2+(3/(1+a/(2-a))))/(a+2b), is there an answer that looks clearer?
 
Jan 2009
454
132
I think I'll come back to this later. I started with how I would approach it.

log(28)/log(35) = log(4*7)/log(5*7) = [log(4) + log(7)] / [log(5) + log(7)] = [2 log(2)/log(7) + 1] / [log(5)/log(7) + 1]

a = log(49) / log(14) = 2 log(7) / log(2*7) = 2 log(7) / [log(2) + log(7)] = 2 / [log(2)/log(7) + 1]. Thus log(2)/log(7) = 2/a - 1
b = log(5) / log(14) = log(5) / log(7*2) = log(5) / [log(7) + log(2)] = 1 / [log(7)/log(5) + log(2)/log(5)]. Thus log(5)/log(7) = 1/b - log(2)/log(5)

log(2)/log(5) = log(2)/log(7) * log(7)/log(5) = (2/a - 1) * (1 / (1/b - log(2)/log(5))
From here, isolate log(2)/log(5). (breakpoint)
 
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Jan 2009
454
132
Isolating log(2)/log(5) was a dead end.
Instead, note that (1/2)a = log(7)/log(14) and b = log(5)/log(14) and hence 2b/a = log(5)/log(7)

We can then use replacement
log(28)/log(35) = [2 log(2)/log(7) + 1] / [log(5)/log(7) + 1]
with
log(2)/log(7) = 2/a - 1 = (2-a)/a
log(5)/log(7) = 2b/a

and you will get the same answer as Idea's above.