if a^x=(a/k)^y=k^m then 1/x_1/y=?
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32
a^x = (a/k)^y = k^m = P ( Let)
a^x = P
take log both sides,
xloga = logP
x = logP/loga ------------(1)
similarly ,
(a/k)^y = P
take log both sides ,
ylog(a/k) = logP
y = logP/log(a/k) ----------(2)
k^m = P
mlogk = logP
m = logP/logk--------(3)
now, 1/x - 1/y = 1/{logP/loga} - 1/{logP/log(a/k}
= loga/logP - log(a/k)/logP
={loga - log(a/k)}/logP
={loga - loga + logk}/logP
= logk/logP
= 1/{logP/logK}
from equation (3)
= 1/m
hence , 1/x - 1/y = 1/m
a^x = P
take log both sides,
xloga = logP
x = logP/loga ------------(1)
similarly ,
(a/k)^y = P
take log both sides ,
ylog(a/k) = logP
y = logP/log(a/k) ----------(2)
k^m = P
mlogk = logP
m = logP/logk--------(3)
now, 1/x - 1/y = 1/{logP/loga} - 1/{logP/log(a/k}
= loga/logP - log(a/k)/logP
={loga - log(a/k)}/logP
={loga - loga + logk}/logP
= logk/logP
= 1/{logP/logK}
from equation (3)
= 1/m
hence , 1/x - 1/y = 1/m
Answered by
35
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![Now, \\ \\ \frac{z {}^{ \frac{1}{x} } }{z {}^{ \frac{1}{y} } } = \frac{a}{ \frac{a}{k} } \: \: \: \: \: \: [ from \: (1)\: \: and \: \: (2)] \\ \\ = > z {}^{ \frac{1}{x} - \frac{1}{y} } = \frac{a}{a} \times k \\ \\ = > z {}^{ \frac{1}{x} - \frac{1}{y} } =k \\ \\ = > z {}^{ \frac{1}{x} - \frac{1}{y} } = z {}^{ \frac{1}{m} } \: \: \: \: \: \: \: \: [from \: \: (3)]\\ \\ = > \frac{1}{x} - \frac{1}{y} = \frac{1}{m} Now, \\ \\ \frac{z {}^{ \frac{1}{x} } }{z {}^{ \frac{1}{y} } } = \frac{a}{ \frac{a}{k} } \: \: \: \: \: \: [ from \: (1)\: \: and \: \: (2)] \\ \\ = > z {}^{ \frac{1}{x} - \frac{1}{y} } = \frac{a}{a} \times k \\ \\ = > z {}^{ \frac{1}{x} - \frac{1}{y} } =k \\ \\ = > z {}^{ \frac{1}{x} - \frac{1}{y} } = z {}^{ \frac{1}{m} } \: \: \: \: \: \: \: \: [from \: \: (3)]\\ \\ = > \frac{1}{x} - \frac{1}{y} = \frac{1}{m}](https://tex.z-dn.net/?f=Now%2C+%5C%5C+%5C%5C+%5Cfrac%7Bz+%7B%7D%5E%7B+%5Cfrac%7B1%7D%7Bx%7D+%7D+%7D%7Bz+%7B%7D%5E%7B+%5Cfrac%7B1%7D%7By%7D+%7D+%7D+%3D+%5Cfrac%7Ba%7D%7B+%5Cfrac%7Ba%7D%7Bk%7D+%7D+%5C%3A+%5C%3A+%5C%3A+%5C%3A+%5C%3A+%5C%3A+%5B+from+%5C%3A+%281%29%5C%3A+%5C%3A+and+%5C%3A+%5C%3A+%282%29%5D+%5C%5C+%5C%5C+%3D+%26gt%3B+z+%7B%7D%5E%7B+%5Cfrac%7B1%7D%7Bx%7D+-+%5Cfrac%7B1%7D%7By%7D+%7D+%3D+%5Cfrac%7Ba%7D%7Ba%7D+%5Ctimes+k+%5C%5C+%5C%5C+%3D+%26gt%3B+z+%7B%7D%5E%7B+%5Cfrac%7B1%7D%7Bx%7D+-+%5Cfrac%7B1%7D%7By%7D+%7D+%3Dk+%5C%5C+%5C%5C+%3D+%26gt%3B+z+%7B%7D%5E%7B+%5Cfrac%7B1%7D%7Bx%7D+-+%5Cfrac%7B1%7D%7By%7D+%7D+%3D+z+%7B%7D%5E%7B+%5Cfrac%7B1%7D%7Bm%7D+%7D+%5C%3A+%5C%3A+%5C%3A+%5C%3A+%5C%3A+%5C%3A+%5C%3A+%5C%3A+%5Bfrom+%5C%3A+%5C%3A+%283%29%5D%5C%5C+%5C%5C+%3D+%26gt%3B+%5Cfrac%7B1%7D%7Bx%7D+-+%5Cfrac%7B1%7D%7By%7D+%3D+%5Cfrac%7B1%7D%7Bm%7D+)
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