Laws of Logarithm

1. a) let P = b^x, then log_bP = x

Q = b^y, then log_bQ = y

PQ = b^x X b^y = b^x+y (laws of indices)

Log_bPQ = x + y

:.             Log_bPQ = log_bP + Log_bQ

1. b) P÷Q = b^x÷by = b^x+y

Log_bP/Q = x –y

:.             Log_bP/Q = logbP – logbQ

1. c) Pⁿ= (b^x)ⁿ = b^xn

Log_bpⁿ = nb^x

:.             LogPⁿ = logbP

1. d) b = b¹

:.             Log_bb = 1

1. e) 1 = b⁰

Logb1 = 0

EXAMPLE – Solve each of the following:.

1. a) Log₃27 + 2log₃9 – log₃54
2. b) Log₃5 – log₃10.5
3. c) Log₂8 +  log₂3
4. d) given that log₁₀2 = 0.3010 log₁₀3 = 0.4771 and log₁₀5 = 0.699 find the log₁₀64 + log₁₀27

Solution

1. a) Log₃27 + 2 log₃9 – log₃54

= log₃ 27 + log₃ 9² –log₃54

= log₃ (27 X 9²/54)

= log₃ (27¹ X 81/54) = log₃ (81/2)

= log₃ 3⁴/log32

= 4log₃ 3 – log₃ 2

4 X (1) – log₃ 2 = 4 – log₃ 2

= 4 – log₃ 2

13. b) log₃ 5 – log₃ 10.5

= log₃ (13.5)- Log310.5 = log₃ (135/105)

= log₃ (27/21) = log₃ 27 – log₃ 21

= log₃ 3 ³ – log₃ (3 X 7)

= 3log₃ 3  – log₃ 3 -log₃7

= 2 – Log₃ 7

1. c) Log₂8 + Log₃3

= log₂2³+ log₃3

= 2log₂2 + log₃3

2+1=3

1. d) log₁₀ 64 + log₁₀ 27

log₁₀ 2⁶ + log₁₀3³

6 log₁₀ 2 + 3 log₁₀ 3

6 (0.3010) + 3(0.4771)

1.806 + 1.4314 = 3.2373.

EVALUATION ( USE THE DISCUSSION BOX AT THE BOTTOM TO SUBMIT YOUR ANSWER FOR DISCUSSION AND APPRAISAL)

1. Change the following index form into logarithmic form.

(a)          6³= 216 (b) 3³ = 1/27        (c) 9² = 81

2. Change the following logarithm form into index form.

(a)          Log₈8 = 1              (b) log _½¼ = 2

3. Simplify the following
4. a) Log₅5 + log52
5. b) ½ log₄8 + log₄32 – log₄2
6. c) Log₃81
7. Given that log 2 = 0.3010, log3 0.477

Log₁₀5 = 0.699, find the log₁₀ 6.25 + log₁₀