Quiz Answers
Indices, Surds & Logarithms
 
 
                       x= lg a
                   10x = a                                 convert to index form
 
                            102x+1 - 7 (10x) = 26
                   10(10x)2 - 7 (10x) - 26 = 0
                          substituting x = lg a,
                              10a2 - 7a - 26 = 0
                         (10a + 13) (a - 2) = 0
 
                    a = -13/10                          rejected as a > 0
     or            a = 2
 
 
                     log2x - 4 logx16 = 0
                                   log2x = 4 logx16
                                   log2x = 4 (log216 / log2x)      change of base
                                           = 16 / log2x
                               (log2x)2 = 16
 
                   log2x = 4                 or                 log2x = -4
                        x = 16                                         x = 1/16
 
 
                       32x+1 - 2 (3x+0.5) - 3 = 0
                3 (3x)2 - 2 (30.5) (3x) - 3 = 0
                       Let y = 3x
                          3y2 - 2 (30.5)y - 3 = 0
                              
                                             
                                   or              rejected as 3x >0
                =
                =
             x = 0.5
 
                           (ln x)2 = ln x2
                                            = 2 ln x
                (ln x)2 - 2 ln x = 0
                  ln x (ln x - 2) = 0
 
                  ln x = 0                 or              ln x = 2
                      x = 1                                    x = e2
 
                       2 (7x-1) = 5x
                 lg (2 (7x-1)) = lg 5x
         lg 2 + (x - 1) lg 7 = x lg 5
        lg 2 + x lg 7 - lg 7 = x lg 5
               x (lg 7 - lg 5) = lg 7 - lg 2
                                              
                                  = 3.72            approximated to 3 significant figures
 
 
             log2a + log2b = 4
                       log2ab = 4
                             ab = 24
                                 = 16
 
                             ab = 42
                        log4ab = 2
 
                             ab = 84/3
                        log8ab = 4/3