Could someone please tell me better method to solve this using logs?

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Answer 1 · 2018-07-12T13:23:43

Kindly refer to a Proof in the Explanation.

Given that, #a^x=b^y=(ab)^(xy)=k, say.#

Now, #a^x=k rArr lna^x=lnk rArr xlna=lnk#.

#:. x=lnk/lna..................(1)#.

Likewise, #b^y=k, and, (ab)^(xy)=k#.

# rArr y=lnk/lnb..............(2), and, xy=lnk/ln(ab)..................(3)#,

Therefore, from #(1) and (2)#, we get,

#1/x+1/y=lna/lnk+lnb/lnk#,

#=(lna+lnb)/lnk#,

#=ln(ab)/lnk#.

# rArr 1/x+1/y=1/(xy).......................................[because, (3)]#.

#:. (x+y)/(xy)-1/(xy)=0#.

#:. 1/(xy)(x+y-1)=0#.

#"Since, "1/(xy)!=0; x+y-1=0, or, x+y=1#, as desired!