Mathematics

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 

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

  ; Always remember for base, ,   (base cant be zero or negative or 1, it can be less than 1 . e.g. 3/2, 1/2) And for negative value log doesn't exist  (why??) Reason is simple we cant get any negative number even not get zero whatever be the power of a positive number. e.g.   ,  positive or negative what do you think ??? clearly positive. multiplying 1/2 , 55 times and get positive values. therefore,  so, whenever you see a logarithmic expression ,  ;  ,   and  Properties: 1.   ;  clearly, whatever be the value of "a",  2.  ;    , 3.   and  ;   4. Base changing formula:-  ( in second term you can take any base) Proof:  consider,   ;     ....................... (i)  (by definition) consider,    ;   let   and   ...................... (ii) since, or,  therefore, from (i) and (ii), 5.   ;   and  6.     ;    and  7.    ;    8.   ;     and   (note) proof is simple left to reader. 9.  take log with base 'b