2.0 NUMERICAL PROCESSES (2) THEORY OF LOGARITHMS & INDICES

   OBJECTIVES

   1. Express statements given in index form (such as 81 = 3^4) as an equivalent logrithms statement (log 3 81 = 4).
   2. Evaluate expression given in logarithms form.
   3. Note the equivalence between the laws of indices and the law of logarithms
   4. Recall and use the law of logarithms to simplify and/or evaluate given expression without the use of logarithm table.
   5. Use logarithms table for the purpose of calculation.

2.1 LAW OF LOGARITHMS & INDICES

   The three fundamental law of indices can be stated in their equivalent logarithms form;

   1. In indices: Y^a x Y^b = Y^a+b (Note ^ means Raise to power)

   * In Logarithms: Log(MN) = Log M + Log N

   2. In indices: Y^a ÷ Y^b = Y^a-b (Note ^ means Raise to power)

   * In Logarithms: Log(M/N) = Log M - Log N

   3. In indices: (X^a)^b = X^ab (Note ^ means Raise to power)

   * In Logarithms: Log(M^p) = p log M



   Example 1: Simplify Log 8 + Log 5

      solution;
          Log 8 + Log 5 = Log(8x5)
             Ans = Log 40

   Example 2: Simplify Log 9 ÷ Log 3

      solution;
         Log 9 ÷ Log 3 = Log 9/Log 3
         = Log 3^2 / Log 3^1 (Note / means All over)
         = 2 Log 3 / 1 Log 3 (Log 3 will cancel Log 3)
          = Log 2/1
          Ans = Log 2

   Example 3: Given that Log 2 = 0.30103; Calculate Log 5 without using table

      solution;
         Log 5 = Log 10/2
         = Log 10 - Log 2 (Note: Log 10 = 1)
         = 1 - 0.30103
         Ans = 0.69897

   Example 4: Evaluate Log base3(6.84) to 2 d.p



   solution;
      Log3 (6.84)
      = Let Log3 (6.84) = x
      then 3^x = 6.84
      = log(3^x) = log(6.84)
      x = log(6.84) / log(3)
      x = 0.8351 / 0.4771
      Ans = 1.75

2.2 CALCULATIONS USING LOGARITHM TABLE

   Example 1: Evaluate 82.47 x 24.85 / 209.3

      solution;
       Draw a table form with "No & Log"

       No 82.47 = log 1.9163 (No means Number)
       No 24.85 = log 1.3954
       Add together = 3.3117

       No 209.3 = log -2.2307 (from log table)
       Deduct -2.2307 from 3.3117

       = 3.3117 - 2.2307
       Ans = 0.9910

2.3 LAW OF INDICES

   The following laws of indices are true for all non-zero value a, b and x

   1. X^a x X^b = X^a+b (Note ^ means Raise to power)

   2. X^a ÷ X^b = X^a-b (Note ^ means Raise to power)

   3. X^0 = 1

   4. X^-1 = 1 / X^a (Note / means All over)

   5. (X^a)^b = X^ab

   6. X^1/a = a√x (Note √ means Square root)

   7. X^a/b = b√x^a or (b√x)^a



2.3.1 WORKED EXAMPLES

   Example 1: Simplify 25^1/2

      solution;
         25^1/2 = √25
         Ans = 5

   Example 2: 4^3 ÷ 4^5

       solution;
          4^3 ÷ 4^5 = 4^3-5
          = 4^2
          = 1 / 4^2
          Ans = 1/16





[1] Simplify 3^8 x 3^3
[2] Simplify 5^3 x 5^-1
[3] Express Log 3 + Log 4
[4] Evaluate 3Log2 + Log20 - Log1.6
[5] Simplify Log 8 - Log 4
[6] Simplify Log 8 ÷ Log 4
[7] Simplify Log 4 / Log 2
[8] Simplify (27/48)^3/2
[9] Simplify 3^6 ÷ 3^2
[10] Simplify (4/25)^-1/2 x (2^4) ÷ (15/2)^-2

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