342. Exponential or logarithmic equations/inequations.
by Virgil Nicula, May 26, 2012, 12:56 AM
PP1. Study the monotony of the function 
, where 
and 
.
Proof. Let
and 
. Thus, 
. Observe that the functions 
are bijective strict increasing 
and the function 
is bijective
strict decreasing
on ![$(0,1]$](//latex.artofproblemsolving.com/1/0/1/101678300d1aa11009468da8419aaf2c57b795bd.png)
and bijective strict increasing on 
. Thus, 
and 
are positive strict decreasing
on and are positive strict increasing on . Therefore, their product 
is positive strict decreasing on and
strict increasing on . Thus, 
i.e. 
, i.e. 
.
Hence
solution of equation 
is 
, where 
and 
.
PP2. Solve the logarithmic equation
, where and 
.
Proof. Define the equivalence relation over
. Since 
, our equation
becomes
. Denote 
. Observe that 
, i.e. 
and 
. But 
. Thus our equation becomes 
, i.e. 
.
, i.e. 
. Thus, 
and 
, i.e. 
, where 
.
Thus,
and 
, i.e. 
. Hence our equation has only the zero 
, where 
and 
.Proof. Let
and 
. Thus, 
. Observe that the functions 
are bijective strict increasing 
and the function 
is bijectivestrict decreasing
on ![$(0,1]$](http://latex.artofproblemsolving.com/1/0/1/101678300d1aa11009468da8419aaf2c57b795bd.png)
and bijective strict increasing on 
. Thus, 
and 
are positive strict decreasing on
and are positive strict increasing on . Therefore, their product 
is positive strict decreasing on andstrict increasing
on . Thus, 
i.e. 
, i.e. 
.Hence
solution of equation 
is 
, where 
and 
.PP2. Solve the logarithmic equation
, where and 
.Proof. Define the equivalence relation over
. Since 
, our equationbecomes
. Denote 
. Observe that 
, i.e. 
and 
. But 
. Thus our equation becomes 
, i.e. 
.
, i.e. 
. Thus, 
and 
, i.e. 
, where 
.Thus,
and 
, i.e. 
. Hence our equation has only the zero 
This post has been edited 28 times. Last edited by Virgil Nicula, Nov 18, 2015, 9:43 AM
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