[AM_20150616LGDFCA] Logarithms and Knowing that You are Correct

Question

Introduction

     This is a relatively straightforward question once the student has learned the rules of logarithms.  When I was first learning logarithms it took me quite some time to get used to the idea of “logs”.  Are they fallen trees?  So what are “logs”?  They are just the exponents or indices.  For example  2³ = 2 ´ 2 ´ 2 = 8  and we can write  log₂8 = 3.  Logarithm to base  2  of  8  is 3, because  3  is the index  i.e. to get  8  you need to multiply  2  by itself  3-fold.

     In general,  log_ba = x   Û   a = b^x.  Why?  Because that is exactly what logarithm means!  One way to remember this definition is to imagine: if you transport the log to the other side of the equation, the log drops off and you get the base  b  propping up the  x.  You can also do it the other way round.  If the base  b  of a power moves to the other side, it becomes a  “log”  with base  b.  [active mnemonics]

     What about the “common logarithm” lg?  It is the logarithm with base  b = 10.  In the days before pocket calculators were prevalent, students used books and slide-rules with logarithms of base  10  for multiplying and dividing large numbers.  Base 10 logarithms are still commonly used in today for the Richter Scale (in seismology, to measure earthquakes), for decibels (to compare the loudness of sounds or gain / loss in amplifiers), for pH (measurement of acidity / alkalinity in Chemistry) .... etc.  The aforementioned rule works exactly the same way, with  b = 10.

                      lg a = x   Û   a = 10^x                 (lg means log₁₀)

Note that in many calculators, their “log” button is for  lg  or logarithm of base 10.

Solution

Checking Your Answer

     The person who posted this question on Facebook got  33 333 333.3  as his answer, but did not realise that his answer is the same as the “model” answer, which is given to three significant figures in standard scientific notation.  Many students have the habit of checking their answers against the “model” answer usually given at the back of the book or worksheet, which may sometimes be wrong!  Anyway, in tests and examinations, you do not have the luxury of checking your answers like this.  In real life, if an engineer makes a calculation mistake, buildings may collapse and people die.  It is better to make it a habit to check your answers on your own and to know and be sure that you are correct.  One way to do this is to substitute the value of  x  back into the original equation to see if it works.  Nowadays, many models of calculators have a “store” function indicated by a button labelled with “STO” or an arrow “®” or something like that.  You can store the value into a variable (or memory location) like  X  and then key in something like  “log(3X) ”  and see whether you get  9  or something close.  Be aware that calculators can have rounding errors. 

Notations for “log”

     School students are taught to use “lg” to mean “log₁₀”  and  “ln”  to mean the natural logarithm “log_e”  where the special number  e = 2.7182818284 ...  discovered by the visually impaired but brilliant mathematician Euler.  Many calculators take “log” to mean “lg”  or  “log₁₀”.  For adult working professionals, “log” (without indication of the base) usually depends on what field they are in, or on the topic being discussed.  As mentioned before, base 10 is used for Richter scale, decibels and pH.  Computer scientists tend to use base  2  because of the binary system.  For rate of reaction (chemistry) or radioactive decay (chemistry / physics), the natural logarithm “ln” is often used.  In school, for the purposes of learning, we make the logarithm bases explicit.  Do not simply write “log”.  Write “lg”, “ln” or “log₂” or “log₇” or “log_b” (for whatever  b  is).  Note also that the letter “l” in all these notations is not the letter “i” or “I”, but it is the smaller case “L” (for logarithms).

Heuristics Used

H05. Work backwards

H09. Restate the problem in another way

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘O’ Levels Additional Mathematics

* International Baccalaureate (IB) Mathematics (revision)

* other syllabuses that involve logarithms and exponentials