A Logarithms Quiz

I gave the following quiz in class recently

If log _b D = 16 and log _b H = 12, simplify:

(a) log _b(DH)                                                                                                                                                

(b) log _b (1/H)                                                                                                                           

(c) log _b√D                                                                                                                 

(d) log _H D.                 

Two students took quite different approaches.  Johnny’s quiz was as follows:

(a) log _b (DH ) = log _b (X ) + log _b (X ) = 16 + 12 = 28

(b) log _b (1/H) =  log _b (H^-1) = - log _b (H) = -12

(c) log _b√D      = log _b D ^0.5         = 0.5 log _b D= 0.5 × 16  = 8

(d) log _H D =    log _b D ÷ log _b H = 16/12 = 1.33

Mary’s quiz was as follows:

log _b D = 16 à D = b ¹⁶     and   log _b H = 12  à H = b ¹²

(a) log _b (DH )

=  log _b (b ¹⁶ b ¹²)

= log _b (b ¹⁶⁺¹²)

= log _b b ²⁸

 = 28

(b) H = b ¹²

1/H = 1/ (b ¹²)

= b^-12

So log _b (1/H)

= log _b b^-12

= 12

(c) D = b ¹⁶     

So √D = √( b ¹⁶)

= b ⁸

So log _b√D

= log _b  b ⁸

= 8

(d) not attempted

Johnny had a perfect score. (In our syllabus, decimal answers are rounded to three significant figures so 4/3 = 1.33 is ‘correct’). Mary made a small mistake, forgetting the minus sign on part (c), and omitting part (d) entirely.   However, it is more difficult to say who has a better understanding of logarithms.  Has Johnny just memorized the rules, while he follows blindly?  Or is he using the rules because class quizzes place a premium on speed and efficiency?   Is Mary a more careful mathematician because she works from first principles?  Or is she using the index laws because she has not done her revision for the quiz and does not actually know the rules of logarithms- which she accordingly needs to derive, ‘on the spot’?