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!