Grade 12Algebra
x(y+z-x)/logx = y(x+z-y)/logy = z(x+y-z)/logz so show that,
xyyx = yzzy = zxxz ................
1 Answer
Arpit Jaiswal
nice question but cracked 
let the given be equal to k-
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x(y+z-x)/logx = y(x+z-y)/logy = z(x+y-z)/logz =k
or, logx=x(y+z-x)/k, multiplying by y on both sides, we get- ylogx=xy(y+z-x)/k ....(1) also, multiplying both sides by z we get- zlogx=zx(y+z-x)/k ........(2)
and, logy=y(x+z-y)/k, multiplying both sides by x, we get- xlogy=xy(x+z-y)/k .......(3) also, multiplying both sides by z we get- zlogy=zy(x+z-y)/k ...........(4)
and, logz=z(x+y-z)/k, multipying both sides by y, we get- ylogz=yz(x+y-z)/k .......(5) also, mulitplying both sides by x we get- xlogz=xz(x+y-z)/k ..........(6)
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adding 1 and 3, we have- log(x^y)(y^x)=xyz/k |
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adding 4 and 5, we have- log(y^z)(z^y)=xyz/k |
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adding 2 and 6, we have- log(z^x)(x^z)=xyz/k comparaing the RHS of above and removing log, we have- (x^y)(y^x)=(y^z)(z^y)=(x^z)(z^x) Hence proved. AJ
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