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If x = loga(bc), y = logb(ca) and z = logc(ab) prove  that

+ y +  xy-  2.

OldTimer  Sep 29, 2017
 #1
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+2

3 important logarithmic identities:

 

1.loga(b)=1/(logb(a))

2. loga(bc)=loga(b)+loga(c)

3. loga(b)*logb(c)=loga(c)

 

x+y+z= loga(bc)+logb(ca)+logc(ab)=(loga(b)+loga(c))+(logb(c)+logb(a))+(logc(a)+logc(b)).

 

d=loga(b)

e=loga(c)

f=logb(c)

 

x+y+z=(loga(b)+loga(c))+(logb(c)+logb(a))+(logc(a)+logc(b))=d+e+f+1/d+1/e+1/f

 

x*y*z-2=(loga(b)+loga(c))*(logb(c)+logb(a))*(logc(a)+logc(b))=(d+e)*(f+1/d)*(1/e*1/f)=(d*f)/e+d+1/e+1/f+f+e+1/d+e/(d*f)-2=d+e+f+1/d+1/e+1/f+((d*f)/e+e/(d*f)-2). 

 

d*f=loga(b)*logb(c)=loga(c)=e therefore e/(f*d)=(d*f)/e=1 meaning:

 

x*y*z-2=d+e+f+1/d+1/e+1/f+((d*f)/e+e/(d*f)-2)=(d+e+f+1/d+1/e+1/f)+((d*f)/e+e/(d*f)-2)=(x+y+z)+(1+1-2)=(x+y+z)+0=x+y+z

 

therefore x+y+z=x*y*z-2

 

~blarney master~

Guest Sep 29, 2017
 #2
avatar+87708 
+2

Note....    logd e  *  loge f =   log f

Also .....     log e / log d   +  log f / log d  =   logd (ef)

Also........ [ log a / log c ] [ log c / log a ]  =   1

Also ...........( logb c *  logc b )  =  1

And.....    logd e + logd f  =  logd (ef)

 

So......

 

xyz  =   [  logb / loga   + log c /  log a ]  [  log c / log b  + log a / log b ] [ log a / log c + log b / log c ]  =

 

[ ( logb / loga)( log c / log b) + (logb / loga)( log a / log b) + ( log c /  log a)  (  log c / log b) +

(log c /  log a) (log c /  log a)]  [ log a / log c + log b / log c ]

 

[ loga c  + 1  +  log c /  loga *  log c / log b  +  logb c ]  [ logc a + logc b  ] =

 

[   loga c  + 1   +  loga c * logb c   +   logb c ]   [ logc a + logc b  ]  =

 

[ ( loga c) ( logc a) + (1)( logc a) + ( logc a * loga c) * logb c + (logb c)(logc a ) + ( loga c)( logc b) + (1)( logc b) + loga c *( logb c *  logc b )  +  ( logb c)(logc b )  ]

 

[  1 +  logc a  + 1* logb c  +  logb a  + loga b +  logc b +   loga c * 1  + 1  ]  =

 

[ 2  + loga b +  loga c +  logb c  +  logb a +   logc a  +  logc b ]  =

 

2   +  loga (bc)  + logb(ca)  + logc (ab)

 

2  +  x  +  y  + z           (  subtract 2 from both sides)

 

xyz  - 2   =  x + y + z

 

 

 

cool cool cool

CPhill  Sep 29, 2017
edited by CPhill  Sep 29, 2017
edited by CPhill  Sep 29, 2017
edited by CPhill  Sep 29, 2017
edited by CPhill  Oct 4, 2017

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