**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;

Log

*3* (6.84)

= Let Log

*3* (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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