I gave the following quiz in class recently

If log

*= 16 and log*_{b}D*12*_{b }H =*,*simplify:
(a) log

*(*_{b}*DH*)
(b) log

*(1/*_{b}*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

*(1/*_{b}*H*) = log*(*_{b}*H*) = - log^{-1}*(*_{b}*H*) = -12
(c) log

_{b}**√***D*= log_{b }D^{0.5}^{ }= 0.5 log_{b}*D= 0.5 × 16 = 8*
(d) log

*= log*_{H}D*og*_{b}D ÷ l_{b }H = 16/12 = 1**.**33
Mary’s quiz was as follows:

log

*= 16 à D = b*_{b}D^{16 }and log*12 à H = b*_{b }H =^{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

*(1/*_{b}*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’?

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