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If $\log_y x = a \cdot \log_z y = b \cdot  \log_x z = ab$ then which of the following pairs of values for $(a,b)$ is not possible?

  1. $-2, 1/2$
  2. $1,1$
  3. $0.4, 2.5$
  4. $\pi, 1/\pi$
  5. $2,2$
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$\log_{y}x$ = a

$\frac{\log  (x)}{\log (y)}$ = a  ---------(P)

$\log_{z}y$ = b 

$\frac{\log(y)}{\log( z)}$ = b  ---------(Q)

If we multiply eqn P and Q the we get,

$\frac{\log (x)}{\log (z)}$ = ab  ---------(R)

and we are given,

$\log_{x}z$ = b 

$\frac{\log (z)}{\log (x)}$ = ab   

from eqn R,

$\frac{1}{ab}$ = ab.

If we put a=2 and b=2 then the above equation does not satisfy where as all other options satisfy the equation.

Ans- 5. (2,2)  

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