Logarithmic expression inequality concepts - III

MOST IMPORTANT CONCEPTS

if,    ,  , 

i) If  ,   

ii) If   , 

why ?????

take  



increasing with increase in power,



log is also increasing, with increase in the value of 'x' ()
means,


and 

means,  is increasing with increase in the value of  "x",  if 

Now, take 



decreasing with increase in power



decreasing with increase in value of x ( )

i.e. 



i.e.  is decreasing with increase in value of "x" if 

Thus,


if,    ,  , 

i) If  ,   

ii) If   , 


By Graph,

we know,

clearly from figure, 


if,    ,  , 

i) If  ,   

ii) If   , 


Example:




first find the values of x for which the inequality is defined.

STEP 1:



(since ,  )

or, 
by wavy curvy, (previous post),

  ........................................... (A)

therefore inequality is defined in above interval,



STEP 2:
now solve the inequality,



since base is greater than 1, log is increasing, we know for any base , 

 

 ( we can also see from graph of , that for x > 1, value of  is grater than zero, therefore for   , )

from wavy-curvy method,

 .......................... (B)

STEP 3:

but the inequality is defined on (A)

therefore, only those solutions are valid or defined which lies in the interval (A)

take the intersection of (A) and (B) ,

we get the solution of inequality,

ANSWER  


RULES/STEPS TO SOLVE INEQUALITY/EQUALITY


STEP 1. First find the values for which inequality/equality is defined (Let the solution be "A")

STEP 2. Solve the inequality/equality ( Let the solution is "B")

STEP 3. After solving the inequality take only those values from the solution which belong to "A" ( STEP 1 INTERVAL/SETS).

                                                          OR

Take the intersection of STEP 1 interval/sets and STEP 2 interval/sets that is