Question

Log base y to x . log base z to y . log base x to z

simplify the above equation and pls answer fast

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{log\,_yx\;.\;log\,_zy\;.\;log\,_xz}$

$\underline{\textbf{To simplify:}}$

$\mathsf{log\,_yx\;.\;log\,_zy\;.\;log\,_xz}$

$\underline{\textbf{Solution:}}$

$\underline{\textbf{Base change formula:}}$

$\boxed{\mathsf{log\,_ab\;log\,_bc=log\,_ac}}$

$\mathsf{Consider,}$

$\mathsf{log\,_yx\;.\;log\,_zy\;.\;log\,_xz}$

$\mathsf{=(log\,_zy\;log\,_yx).\;log\,_xz}$

$\mathsf{=(log\,_zx).\;log\,_xz}$  $\textsf{(Using base change formula)}$

$\mathsf{=log\,_zz}$  $\textsf{(Using base change formula)}$

$\mathsf{=1}$

$\implies\boxed{\mathsf{log\,_yx\;.\;log\,_zy\;.\;log\,_xz=1}}$

Answer 2

Concept:

To solve this question, We need to recall the following formula of logarithm function:

(1)  $log_{a} b=\frac {log_{e}b}{log_{e}a}$

Given:

The expression : $log_{y}x\;.\;log_{z}y\;.\;log_{x}z$

To find:

The simplified form of the given exprression.

Solution:

The given expression is:

   $log_{y}x\;.\;log_{z}y\;.\;log_{x}z$

By using the formula we can rewrite this in the form:

$=\frac {log_{e}x}{log_{e}y}\;.\;\frac {log_{e}y}{log_{e}z}\;.\;\frac {log_{e}z}{log_{e}x}$

On cancelling the common terms of numerator and denominator we get:

= 1

Hence, the simplified form of equation: $log_{y}x\;.\;log_{z}y\;.\;log_{x}z$ = 1.