Question

IIT advanced problem on Logarithms

Answer 1

I am half dead working on this problem by now

Answer 2

which of those is given and which needs to be proved ?

Answer 3

b^logx=a^logy -> is given

Answer 4

the first thing that comes to mind is
\(\Large \log_ba = \log_yx\)
let me work on paper to see whether this is useful...

Answer 5

It might not be i think ...

Answer 6

Did u delete the reply

Answer 7

what is the exact problem ?
are you trying to prove first statement given the second statement ?

Answer 8

Sorry not prove find x and y

Answer 9

@sidsiddhartha

Answer 10

so its a simultaneous equations problem ?

Answer 11

How will you solve that simultaneously

Answer 12

This problem is a torture

Answer 13

im asking you to clarify what exactly is the problem

Answer 14

we're given two equations and we need to solve x and y
is that right ?

Answer 15

\[\huge (ax)^{\log a } = (by)^{\log b }\]
Given:-
\[\huge b ^{\log x } = a ^{\log y }\]

Question:-
----------> Find x and y in terms of a and b

Answer 16

thank you, im getting x = y
lemme grab a paper :)

Answer 17

wait a minute im doing it:)

Answer 18

with some work, i am getting

a= b (then x=y)
OR
ab =1 (which is not possible)
OR
yb = 1 (which also means ax =1)
so for this y =1/b, x= 1/a
and we can see that this can be one of the solution

Answer 19

wait let me check

Answer 20

Thanks dude

Answer 21

if a,b and x,y are unequal, then 1st case is also not possible

Answer 22

whats the ans is it x=1/a and y=1/b

Answer 23

yes

Answer 24

wait im uploading a photo

Answer 25

\(\log a(\log a+\log x) = \log b (\log b +\log y) \)

dividing both sides by log ab
and using
\(\log_ax = \log_by\)
we will get those 3 conditions

Answer 26

ok

Answer 27

sorry, divide both sides by log a log b

Answer 28

\(\Large \log_ba [1+\log_by] = [1+\log_by]\)

since a cannot be = b
then we have
\(\log_by=-1\)
which gives the required result

Answer 29

THANK YOU ALL OF YOU GUYS

Answer 30

welcome ^_^

Answer 31

:)