How do you prove (log_(a)x)(log_(x)a)=1?

May 6, 2016

Explanation:

Let ${\log}_{a} x = u$

hence $x = {a}^{u}$ ............(A)

and let ${\log}_{x} a = v$,

hence $a = {x}^{v}$ ............(B)

Putting value of $a$ from (B) in (A), we get

$x = {\left({x}^{v}\right)}^{u} = {x}^{v \times u} = {x}^{u \times v}$

or ${x}^{1} = {x}^{u \times v}$

or $u v = 1$

Now putting back values of $u$ and $v$

${\log}_{a} x \times {\log}_{x} a = 1$