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

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Answer 1 · 2016-05-06T17:30:20

Please see below.

Let #log_ax=u#

hence #x=a^u# ............(A)

and let #log_xa=v#,

hence #a=x^v# ............(B)

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

#x=(x^v)^u=x^(vxxu)=x^(uxxv)#

or #x^1=x^(uxxv)#

or #uv=1#

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

#log_ax xx log_xa=1#