Problems based on Exponential function and Power Function

1. POLYNOMIAL :- 

 - Variable
 - Constant

e.g. , 

2. EXPONENTIAL :- 

 - Constant
 - Variable

e.g. , 

3. POWER FUNCTION:-


 - Variable
 - Variable

e.g. 

CONDITION FOR POWER FUNCTION:



if,  , y is a power function
then,
i) , 
ii)  must be defined

e.g.
1.

First step is to find for which points equality is defined

We know log is defined when ( http://iitmathseasy.blogspot.in/2013/03/logarithamic-expression-concepts-ii.html)

 is defined when
a) 
or, 

 is defined when,
b) 
or, 

from definition of power function
c) , 

and,  ( and  )

for x = 0 and 2; |x - 1| = 1 (from graph), 
therefore, 

Remember this NOTE:- Euclid theorem is also important on solving the question of power function.
Recall " Equal things are equal to each other "

d) Since L.H.S. is positive 
means LHS>0
=> RHS>0
means, 

or, 

so equality is defined on,
Intersection of  a) ,b), c) and d) 

.............................................................. (i)

or, 

Now second step is to solve the equality,

we know how to solve modulus equality,
but since equality is not defined for 
we left them,
for 



therefore,



or, 

or,

let, 

therefore,



or,

or,

or,

   or   


here we get the value which is less than 1

THIRD STEP: 
Not possible because not in the interval for which the equality is defined (see equation (i) )

Next, for 



THIRD STEP:
which is in the interval for which the equality is defined see (i)

therefore , ANSWER 






2. 

First step is to find for which points equality is defined,

from the definition of power function,

a) 

and, 

since LHS is positive, power of any positive number is positive,
therefore, RHS > 0( Euclid Theorem)

b) 

therefore take the intersection of a) and b)
the equality is defined in,

 ................................................................ (i)

Second Step to solve the equality,
take log of both side,



or, 

or,  ( take )

 or 

When, 



THIRD STEP: which is in (i) hence valid solution

Next when, 



THIRD STEP:

which is also a valid solution because belongs to (i)

therefore solution is