Posted at 8:29 pm by Sergeant Loi in A-Maths Notes & Tips
月
曜
日
2008
Mon
28
Apr
Sergeant Loi avatar

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

(11)
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. y = a^x doubleleftright log_a{x}
  2. log_a{a} = 1
  3. log_a{1} = 0
  4. lg{x} = log_10{x} → This is what the log key does on your calculator!
  5. ln{x} = log_e{x} → This is what the ln key does on your calculator!

  6. From (1) above, you can deduce that:
    lg{x} = y doubleleftright y = 10^x and ln{x} = y doubleleftright y = e^x

    From (2) 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!*

がんばって!!!
Sergeant Loi

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Comments

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11 Comments

  1. 1
    月
    曜
    日
    2008
    Mon
    28
    Apr
    9:55PM
    PS ignorantsoup says,

    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. 2
    火
    曜
    日
    2008
    Tue
    29
    Apr
    12:13AM
    Sergeant Loi avatar Sergeant Loi says,

    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. 3
    火
    曜
    日
    2008
    Tue
    29
    Apr
    5:46PM
    |c3^sNoW says,

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

  4. 4
    火
    曜
    日
    2008
    Tue
    29
    Apr
    5:54PM
    |c3^sNoW says,

    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. 5
    水
    曜
    日
    2008
    Wed
    30
    Apr
    9:51AM
    Sergeant Loi avatar Sergeant Loi says,

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

  6. 6
    水
    曜
    日
    2008
    Wed
    30
    Apr
    1:52PM
    PS ignorantsoup says,

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

  7. 7
    木
    曜
    日
    2008
    Thu
    1
    May
    12:35AM
    Sergeant Loi avatar Sergeant Loi says,

    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. 8
    木
    曜
    日
    2008
    Thu
    1
    May
    11:03AM
    PS ignorantsoup says,

    :( omg..tickle me then..

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

  9. 9
    木
    曜
    日
    2008
    Thu
    1
    May
    2:22PM
    tohkiatshengKEN says,

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

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

  10. 10
    金
    曜
    日
    2008
    Fri
    2
    May
    11:18AM
    Sergeant Loi avatar Sergeant Loi says,

    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. 11
    水
    曜
    日
    2008
    Wed
    7
    May
    5:13PM
    tohkiatshengKEN says,

    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

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