How many number of solutions of log[5](x-1)=log[2](x-3)?(jee)

there are various methods by which you can solve this problem,

i''ll be following the objective approach as you are mainly concerned for jee,

the fastest method for solving this problem is to consider the graph of logarithmic function!!

yes you got it a DECREASING curve streching from - infinity to +infinity

now for log₅ (x-1) at x=2 we get the curve crossing the X axis

similary for x=4 we get the other curve crossing the X axis

now if you consider the poitn x=6

log₅ (x-1) = 1

and log₂ (x-1) = log₂ 5 which is definitly greater than 1

if you move forward log₂x > log₅x

so the curve will never intersect again

CLEARLY THEIR EXIST ONLY ONE SOLUTION!! that is between 2 and 6

ALTHOUGH THIS APPROACH MAY SEEM TO YOU RATHER ILLOGICAL

BUT BEILIVE ITS THE FASTEST!!!

Approved

Plot individual graphs of RHS and LHS, number of point of intersection represent the number of solution of the equation,

Best of luck!

May god bless you!

1

x=11 only satisfies the condition.
hence only one value is possible