2008
Mon
28
Apr
Sergeant Loi avatar

Sergeant Loi’s Mid-Year Boot Camp 2008 – Fall In For Logarithm Training!

(13)
Tuition given in the topic of A-Maths Tuition Notes & Tips from the tuition centre desk of Sergeant Loi at 8:29 pm (Singapore time)
Log PT

Drop that log and Sergeant Loi will tie you
to a log (and tickle your bare feet)!

*Puts on helmet*

*SHOWS STERN & MEAN FACE*

*Sounds bugle* EVERYBODY FALL IN!

Today’s drill is going to be a little longer (and tougher) than the previous one, since so many of you were tragically killed in action at logarithmic minefields over the years.

So to prevent yourself from adding to the end-of-year body count, get ready for Log PT!

PICK UP YOUR LOGS NOW! EXERCIIIIIIISE BEGIN!

*Cracks whip*

The Laws of Logarithms

A. IF YOU DON’T KNOW THESE, CAN CLOSE SHOP & FORGET ABOUT THE REST!

  1. For log_a{y} to be defined y > 0 AND a > 0

    *This property is often used to select possible values of y when a logarithmic equation has more than one solution.
  2. y = a^x doubleleftright log_a{y} = x
  3. log_a{a} = 1
  4. log_a{1} = 0
  5. lg{x} = log_10{x} → This is what the log key does on your calculator!
  6. ln{x} = log_e{x} → This is what the ln key does on your calculator!
  7. From (2) above, you can deduce that:
    lg{y} = x doubleleftright y = 10^x and ln{y} = x doubleleftright y = e^x

    From (3) above, you can deduce that:
    lg{10} = 1 and ln{e} = 1 → many forget this!

B. THE OFT-MISUNDERSTOOD LAWS OF LOGARITHMS

  1. PRODUCT LAW: log_a(xy) = log_a{x} + log_a{y}
    Attention! DON’T ever do these:
    log_a(xy) = log_a{x}log_a{y} → WRONG!
    log_a(x+y) = log_a{x} + log_a{y} → WRONG!
  2. QUOTIENT LAW: log_a(x/y) = log_a{x} - log_a{y}
    Attention! DON’T ever do these:
    log_a(x-y) = log_a{x} - log_a{y} → WRONG!
    {log_a{x}}/{log_a{y}} = log_a(x-y) → WRONG!
    {log_a{x}}/{log_a{y}} = log_a(x/y) → WRONG!
  3. POWER LAW: log_a{x^r} = {r}log_a{x}
    Attention! DON’T ever do this: (log_a{x})^r = {r}log_a{x} → WRONG! (note the brackets)
  4. CHANGE-OF-BASE LAW: log_a{b} = {log_c{b}}/{log_c{a}} = 1/{log_b{a}}
    Attention! DON’T ever do this: {log_c{b}}/{log_c{a}} = b/a → WRONG!
    e.g. {log_2{64}}/{log_2{4}}64/4!
    Instead it should be {log_2{2^6}}/{log_2{2^2}} = {6log_2{2}}/{2log_2{2}} = 3

SAMPLE PRACTICE QUESTIONS

  1. Solve log_sqrt{5}(2-x) = log_5(5-4x).

    Ans:

    {log_5(2-x)}/{log_5{sqrt{5}}} = log_5(5-4x)
    → using Change-of-Base Law from B(4) above (remember that c can be any arbitrary value)
    doubleright {log_5(2-x)}/{log_5{5^{1/2}}} = log_5(5-4x)
    doubleright {log_5(2-x)}/{1/2} = log_5(5-4x)
    → using Power Law from B(1) and from A(2) the denominator log_5{5} = 1.
    doubleright 2 log_5(2-x) = log_5(5-4x).
    doubleright log_5(2-x)^2 = log_5(5-4x)
    → using Power Law from B(1) again.
    doubleright (2-x)^2 = 5-4x
    → CAN cancel both sides since both logs are of the same base.
    … the rest you should be able to solve on your own!

  2. The mass, m of a radioactive substance, at time t days after being first observed, is given by the formula m = 24e^{-0.02t}. Find the value of t when the mass is half of its value at t = 0.

    Ans:

    When t = 0, m = 24e^{-0.02(0)} = 24 → somehow many fail to realize this

    When m is half of m when t = 0,

    doubleright 24/2 = 24e^{-0.02t}
    doubleright 12 = 24e^{-0.02t}
    doubleright 1/2 = e^{-0.02t}
    doubleright ln {1/2} = ln e^{-0.02t}
    ln both sides when you’re typically looking for a x in ex
    doubleright ln 1 - ln 2 = -0.02t ln e
    → Using Quotient Law from B(2) for L.H.S. and Power Law from B(3) for R.H.S.
    doubleright -ln 2 = -0.02t
    → since ln 1 = 0 from A(3) and ln e = 1 from A(6) above
    doubleright t = {-ln 2}/-0.02 = 34.7

As usual, get these rules drilled into your head! Spot the pointers and common mistakes in red! Understand the representative sample questions! Check out this question for further practice!

Print this out if necessary and stay on the right side of the laws above – for if you don’t Sergeant Loi will literally tie you to a log and unleash unspeakable terrors (e.g. tickling your bare feet) upon you!

*Cracks whip!*

頑張って!!!

Revision Exercise

To show that you have understood what Miss Loi just taught you from the centre, you must:

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Comments & Reactions

13 Comments

  1. PS ignorantsoup's Avatar
    PS ignorantsoup commented in tuition class


    2008
    Apr
    28
    Mon
    9:55pm
     
    1

    Eh you all better listen up and do the logs as sergeant Loi say ok? If you all fail your IPPT this friday I will give sergeant Loi 3 extra duties.

    Platoon Sergeant Ignorantsoup
    (Eh I really PS)

  2. Sergeant Loi's Avatar
    Sergeant Loi commented in tuition class


    2008
    Apr
    29
    Tue
    12:13am
     
    2

    Huh IPPT? This is a mid-year exam boot camp - where extra duties is too lenient a punishment compared to being tickled alive while tied to a log!

  3. |c3^sNoW's Avatar
    |c3^sNoW commented in tuition class


    2008
    Apr
    29
    Tue
    5:46pm
     
    3

    heng ah i used 2 b nt bad @ logarithms. wahaha. no tickling alive 4 me! 😀

  4. |c3^sNoW's Avatar
    |c3^sNoW commented in tuition class


    2008
    Apr
    29
    Tue
    5:54pm
     
    4

    Miss Loi! a fren of a fren of mine got tis pri 5 maths question...if ABC+ACB=CBA, what is A, B ,C?
    waaahh pri sch maths qn so weird these days. :s

  5. Sergeant Loi's Avatar
    Sergeant Loi commented in tuition class


    2008
    Apr
    30
    Wed
    9:51am
     
    5

    ABC+ACB=CBA?! Didn't know there are other Sergeant Lois around tormenting primary school kids with Matrices-like questions these days!

  6. PS ignorantsoup's Avatar
    PS ignorantsoup commented in tuition class


    2008
    Apr
    30
    Wed
    1:52pm
     
    6

    Ah Sergeant Loi, I've got other sergeants like you tormenting kids ma..One PS can have up to 4 sergeants. Haha

  7. Sergeant Loi's Avatar
    Sergeant Loi commented in tuition class


    2008
    May
    1
    Thu
    12:35am
     
    7

    PS Soupie: Sergeant Loi only serves one master - the holy High Priestess! How dare you suggest otherwise!

    Now take off your shoes and prepared to be tickled!

  8. PS ignorantsoup's Avatar
    PS ignorantsoup commented in tuition class


    2008
    May
    1
    Thu
    11:03am
     
    8

    🙁 omg..tickle me then..

    *takes off shoes and cries whilst looking at manly sergeant loi*

  9. tohkiatshengKEN's Avatar
    tohkiatshengKEN commented in tuition class


    2008
    May
    1
    Thu
    2:22pm
     
    9

    haha..
    hmmm...
    EHEH. *poke*
    can set sec 2 one anot shiol?!
    (=

    saya pergi kolong blok one korner relac relac sup rokok shiol...

  10. Sergeant Loi's Avatar
    Sergeant Loi commented in tuition class


    2008
    May
    2
    Fri
    11:18am
     
    10

    MR TOH KIAT SHENG, too little homework for you these days isit? Sec 2 will be covered under E-Maths notes (akan datang)

    P.S. Rokok? NO SMOKING IN CAMP!

  11. tohkiatshengKEN's Avatar
    tohkiatshengKEN commented in tuition class


    2008
    May
    7
    Wed
    5:13pm
     
    11

    WHOAH SIAH LAH...
    pai seh shiol, abang...
    saya malayu lah bang,
    sparking off racial riot now issit?
    siah lah, deprive me of my malboro shiol...
    )=

    tmr headline news come out liao.(:

    CHINESE tuition TEACHER DEPRIVE MALAY BOY! of his malboro. =D

  12. yantho's Avatar
    yantho commented in tuition class


    2008
    Oct
    4
    Sat
    12:29pm
     
    12

    miss loi... i think under ur Amath notes and tips about the logarithm part, there is some mistakes. u said y=lgx -> y =10^x .. i tink it sld be x = 10^y.

  13. Miss Loi's Avatar
    Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class


    2008
    Oct
    5
    Sun
    12:29am
     
    13

    Oh dear. Thanks yantho for pointing out the typo. This and the ln y = x part next to it have been fixed.

    Sergeant Loi is going to get military court marshalled for this!

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