011-0MWALA_LEARN-ANTILOGARITHM-TABLE

Objectives: 011-0MWALA_LEARN-ANTILOGARITHM-TABLE

Antilogarithm Table

Antilogarithm Table (0.00 → 0.99)

This table gives antilogs of decimal parts (mantissas).
Kiswahili: Meza hii inaonyesha antilog ya sehemu ya desimali (mantissa).

Antilogarithms of 10x
Row 01234 56789 Mean1Mean2Mean3 Mean4Mean5Mean6 Mean7Mean8Mean9

💡 How to Use: Take mantissa (decimal part of log), find row & column, then read the antilog value.
Kiswahili: Chukua sehemu ya desimali (mantissa) ya log, tafuta kwenye safu na kolamu, halafu soma thamani ya antilog.

Logarithm & Antilogarithm Notes

Understanding Logarithms (LOG) and Antilogarithms (ANTILOG)

These notes are written in a simple way so that even Form Two students can understand step by step. We shall mix English + Kiswahili for deeper clarity.

1. What is a Logarithm?

A logarithm answers the question: "To what power must I raise 10, to get this number?"
Kiswahili: Logarithm ni swali kwamba "nitaipandisha 10 kwa nguvu gani, ili nipate namba hii?"

Example:
log10(100) = 2 because 102 = 100.
log10(1000) = 3 because 103 = 1000.

2. Structure of a Logarithm

Every logarithm has two parts:

  • Characteristic → The whole number part (integer). Inapatikana kwa kuhesabu idadi ya tarakimu kabla ya decimal point.
  • Mantissa → The decimal part (fraction). Hii tunapata kutoka kwenye log table.

Example:
log10(234.5) ≈ 2.3701
Here:
• Characteristic = 2 (because 234 has 3 digits → 10² < 234 < 10³)
• Mantissa = 0.3701 (this part tunapata kwenye log table)

3. How Logarithm Table is Built

Log tables are built using the formula:

log10(x) = Mantissa + Characteristic

• The mantissa values are calculated by calculator or computer using:
log10(1.00), log10(1.01), log10(1.02)... up to log10(9.99).
• Watafiti waliweka matokeo haya kwenye meza ili tuweze kutumia bila calculator.

Real Life Example:
Mwalimu anataka kujua log10(56).
56 = 5.6 × 10¹ → Characteristic = 1.
Mantissa ya 5.6 inapatikana kwa log table (≈ 0.7482).
Hivyo log(56) ≈ 1.7482.

4. How to Use the Logarithm Table

Steps:

  1. Andika namba yako katika form ya N × 10ᵏ ambapo 1 ≤ N < 10.
  2. Characteristic = k.
  3. Tafuta Mantissa ya N kwenye log table.
  4. Unganisha → log = Characteristic + Mantissa.

Example:
Find log(325).
325 = 3.25 × 10² → Characteristic = 2.
Mantissa ya 3.25 ≈ 0.5119 (kutoka log table).
Hivyo log(325) ≈ 2.5119.

Understanding Antilogarithms (Antilog)

1. What is an Antilog?

Antilog is the reverse of log. It answers the question: "What number has this logarithm?"
Kiswahili: Antilog ni kinyume cha log. Inatuuliza: "Namba gani ina logarithm hii?"

Example:
If log(N) = 2.3010 → then N = antilog(2.3010).
Kwa calculator utapata N ≈ 200.

2. Structure of Antilog

Antilog table usually provides values for the Mantissa part. The Characteristic part tells you where to put the decimal point.

Example:
Antilog(1.2345).
• Mantissa = 0.2345 → angalia kwenye Antilog Table ≈ 1.714.
• Characteristic = 1 → kwa sababu ni "10¹" tunahamisha decimal → 17.14.
Hivyo Antilog(1.2345) ≈ 17.14.

3. Formula for Antilogarithm

If log10(N) = C + M → then N = antilog(C + M) = (antilog(M)) × 10ᶜ

4. Steps to Use Antilog Table

  1. Separate the log into Characteristic (C) and Mantissa (M).
  2. Tafuta antilog ya Mantissa (M) kwenye table.
  3. Multiply result by 10ᶜ (hii inahakikisha decimal ipo sehemu sahihi).
Example:
log(N) = 3.4771.
• C = 3, M = 0.4771.
• Antilog(0.4771) ≈ 3.000.
• N = 3.000 × 10³ = 3000.
Hivyo N ≈ 3000.

🛠 How to Prepare Log & Antilog Table Yourself

Even bila calculator unaweza kufuata idea hizi:

  1. Chagua namba N (mfano 1.01, 1.02, 1.03...)
  2. Tumia formula log10(N) = ln(N) / ln(10).
    (ln = natural log, kwenye calculator au computer huwezi kuepuka hapa wakati wa kuunda table mpya)
  3. Andika matokeo (decimal part = Mantissa) kwenye meza.
  4. Fanya hivi kwa N zote hadi 9.99.
  5. Kwa Antilog table, weka N = 10ᴹ kwa kila Mantissa.
Real Life Example:
Watafiti kabla ya calculator walihesabu hizi values moja moja kwa mikono au mashine rahisi (logarithm machines). Leo unaweza kutumia Excel / Computer kuunda log table yako binafsi.

Reference Book: N/A

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