IIT JEE , AIEEE Forum - Mathematics , Algebra

sankydreams

Which of the following is are true? You may use f(x) = [log(logx)] / logx

A) (log2.1)^log2.2> (log2.2)^log2.1

B) (log4)^log5<(log5)^log4

C) (log30)^log31> (log31)^log30

D)(log28)^log30<(log30)^log28

Rates assured.....plz dont tell me to use scientific calculator and take out the values, i wont be getting it during exams....

sankydreams

kya hua? cud nobody solve this?
i m noticing these days that the number of people solving ques from community is going down
even experts r going low................wats happening?
is this ship sinking?

spideyunlimited

ill try to think of it. i think ive done such questions

nadeemoidu

Does log mean log to base e?

sankydreams

yes log does indeed means log to base e....

nadeemoidu

f(x) = [log(logx)] / logx
f'(x) = [1-log(logx)]/x(logx)²

If f'(x)>0, then f(x) is increasing.

f'(x)>0 if log(logx)<1
if logx<e
if x<e^e
e^e is approximately 15

So for all x< 15 the function [log(logx)] / logx is increasing and after that it is decreasing.

So if x> y and x,y<15 then log(logx) /logx > log(logy)/logy

=> logy log(logx) > logx log (log y)
=> log(logx^logy) > log(logy^logx)
=> (logx^logy) > (logy^logx)

So A is false and B is true.

So if x> y and x,y>15 then log(logx) /logx < log(logy)/logy

=> logy log(logx) < logx log (log y)
=> log(logx^logy) < log(logy^logx)
=> (logx^logy) < (logy^logx)

So C is true and D is false.

So A and C are true.

iitkgp_bipin

Great work Nadeemoidu.

Find the intervals in which f(x) is increasing and decreasing :

If f(x) is increasing in a certain interval then for x>a, f(x) > f(a).
If f(x) is decreasing in a certain interval then for x>a, f(x) < f(a).

Apply these inequalities to the numbers and obtain the result.

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