Finding value!

Algebra Level 2

What is the value of 1 l o g x y x y z + 1 l o g x z x y z + 1 l o g z y x y z \displaystyle{ \frac{1}{log_{xy} {xyz}} + \frac{1}{log_{xz} {xyz}} + \frac{1}{log_{zy} {xyz}} }


The answer is 2.

Abhishek Singh by Abhishek Singh , 24, India

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4 solutions

Kshitij Johary
Jul 17, 2014

We know that, 1 log ⁡ a b = log ⁡ b a \frac { 1 }{ \log _{ a }{ b } } =\log _{ b }{ a } ⇒ 1 log ⁡ x y x y z + 1 log ⁡ y z x y z + 1 log ⁡ x z x y z = log ⁡ x y z x y + log ⁡ x y z y z + log ⁡ x y z x z \Rightarrow \frac { 1 }{ \log _{ xy }{ xyz } } +\frac { 1 }{ \log _{ yz }{ xyz } } +\frac { 1 }{ \log _{ xz }{ xyz } } =\log _{ xyz }{ xy } +\log _{ xyz }{ yz } +\log _{ xyz }{ xz } ⇒ log ⁡ x y z ( x y z ) 2 = 2 \Rightarrow \log _{ xyz }{ { (xyz) }^{ 2 } } =2

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Abhishek Singh
Mar 22, 2014

Please do reshare my problems !

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Please, allow me to nitpick.

You formatting is poorest.

See how

log ⁡ x y ( x y z ) \log_{xy}\left(xyz\right)

is better (and clearer) than

l o g x y x y z log_{xy}^{xyz}

The problem is okay, but the formatting is terrible.

Just to show you the code I used:

"(\log_{xy}\left(xyz\right)")

Just change " to backslash.

Bernardo Sulzbach - 6 years, 11 months ago

But you please folow me !!!!!!!!!!!!!!1

Abhisek Mohanty - 6 years, 2 months ago
Bapu Sethi
Mar 28, 2014

1/logX base Y ---------> logY base X and log X + log Y -----> log XY

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Sharad Gaikwad
Mar 20, 2014

Its simple! By change of base formula the expression is equal to (log xy+log yz+log zx)/ log (xyz). and which is equal to log [(xyz)^2]/ log (xyz)= 2log(xyz)/log(xyz)=2

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Good And simple,thanks K.K.GARG,India

Krishna Garg - 6 years, 11 months ago

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