0064-0MWALA_LEARN-LOGARITHMS-SOLVING-FOREVER-2

Objectives: 0064-0MWALA_LEARN-LOGARITHMS-SOLVING-FOREVER-2

Comprehensive Logarithms Notes

Comprehensive Logarithms Notes / Ufafanuzi Kamili wa Logarithms

1. History of Logarithms / Historia ya Logarithms

Logarithms were introduced by John Napier (1550–1617) to simplify tedious calculations in astronomy and navigation. He discovered a method to convert multiplication and division into addition and subtraction. The method is important historically because before calculators, this reduced errors in large computations. Logarithm tables were later compiled by Henry Briggs.

Kwa Kiswahili: Logarithms zilianzishwa na John Napier ili kurahisisha mahesabu magumu kama ya astonomia na urambazaji (navigation). Walipata njia ya kubadilisha kuzidisha na kugawanya kuwa kuongeza na kutoa (addition & subtraction).

2. Definition / Ufafanuzi

A logarithm answers the question: "To what power must a base be raised to produce a given number?"

Definition:
logb y = x means bx = y

  • b = base (>0, bβ‰ 1)
  • y = argument (positive)
  • x = exponent (power)

Example: log2 8 = 3 because 2Β³ = 8.

3. Fundamental Formulas / Kanuni Muhimu

  • Product Rule: logb MN = logbM + logbN
  • Quotient Rule: logb (M/N) = logbM βˆ’ logbN
  • Power Rule: logb (M^p) = p logbM
  • Change of Base: logbM = logcM / logcb
  • Special Values: logbb = 1, logb1 = 0

Example: log10 1000 = log10 (10Β³) = 3 log1010 = 3.

4. Derivation of Formulas / Ufafanuzi wa Kanuni

Take logb(MN):

  • Let logbM = x β†’ b^x = M
  • Let logbN = y β†’ b^y = N
  • Multiply: MN = b^x Γ— b^y = b^(x+y)
  • Take logb: logb(MN) = logbb^(x+y) = x + y = logbM + logbN

Vivid Example: If a population doubles every year, the growth over 3 years can be expressed using log: log(2Γ—2Γ—2) = log 2 + log 2 + log 2 = 3 log 2.

5. Common Symbols / Alama Zilizotumika

SymbolMeaning / Maana
bBase of the logarithm / Msingi wa logarithm
xExponent / Kiwango cha nguvu
yArgument / Nambari inayohesabiwa
logLogarithm function / Kazi ya logarithm
lnNatural logarithm (base e) / Logarithm ya asili

6. Logarithm Tables / Historia na Method ya Tabular

Before calculators, people used logarithm tables (Napier, Briggs) to compute products and divisions. Method: Convert numbers to log β†’ add/subtract β†’ use antilog to get result.

Step Table Example:

OperationNumberlogSum/DifferenceAntilog
Multiply6 Γ— 15log 6 β‰ˆ 0.7782, log 15 β‰ˆ 1.17610.7782 + 1.1761 = 1.9543antilog(1.9543) β‰ˆ 90

Advice / Ushauri: Always check the number of digits and decimal places when using tables. Mistakes occur in reading or interpolation. Practice helps accuracy.

7. Multiple Solving Methods / Njia Nyingi za Ku Solve

  • Use direct exponential conversion: x = b^y
  • Use properties of logs: add/subtract/multiply exponents
  • Use log tables for manual calculation
  • Use calculators for verification

Example: Solve log x + log (x-2) = 1

  • Method 1: log(a) + log(b) = log(ab) β†’ log(x(x-2)) = 1 β†’ x(x-2) = 10 β†’ xΒ²-2x-10=0 β†’ Solve quadratic.
  • Method 2: trial and error using log tables β†’ approximate values to satisfy equation.

8. Real-Life Applications / Matumizi Halisi

  • Finance: compound interest, growth rates
  • Science: pH in chemistry (log[H+]), decibels in sound
  • Engineering: signal strength, population modeling
  • Navigation & Astronomy: originally for simplifying complex multiplications/divisions
  • Data Analysis: logarithmic scales for plotting wide-range data

9. Formulas Summary / Muhtasari wa Formulas

  • logb MN = logbM + logbN
  • logb(M/N) = logbM βˆ’ logbN
  • logb(M^p) = p logbM
  • Change of base: logbM = logcM / logcb
  • ln(x) = logex
  • Special: logbb =1, logb1 =0

10. Common Mistakes / Makosa ya Kawaida

  • Using negative numbers inside log (argument must be positive)
  • Mixing base and argument incorrectly
  • Forgetting to apply exponent rules
  • Interpolation errors when using log tables

11. Profitable Ideas / Njia za Kufanya Pesa

  • Create a full interactive log calculator app for students
  • Publish solved logarithm worksheets for exam prep
  • Make tutorial videos explaining multiple solution methods
  • Offer consultancy in science & finance for log-based computations
Logarithm & Antilog Table Generator

Logarithm & Antilogarithm Table Formula and Sample / Kanuni na Jedwali

1. Formula to Generate Logarithms

To generate the logarithm of a number N with 4 significant figures, use the formula:

Step 1: Normalize number
N = a Γ— 10^n, where 1 ≀ a < 10, n ∈ β„€

Step 2: Compute log10N using log property:
log10N = log10a + n

Where:

  • N = Number to find log of
  • a = Mantissa part (decimal between 1 and 10)
  • n = Characteristic (integer part of log)

Step 3: Mantissa Approximation for 4-Figure Tables

Using Taylor expansion or linear interpolation: log(1 + d) β‰ˆ 0.4343 Γ— d - 0.216 Γ— d^2 + ..., for small d (d = a-1)

Example: log 2 Normalize: 2 = 2 Γ— 10^0 β†’ a = 2, n = 0 Mantissa: d = 2-1 = 1 β†’ log 2 β‰ˆ 0.4343 Γ— 1 - 0.216 Γ— 1^2 = 0.2183 (approx) Characteristic: 0 β†’ log 2 β‰ˆ 0.3010 (matches table)

2. Formula for Antilogarithms

Given log N = L, find N using:

N = 10^L = 10^(characteristic) Γ— 10^(mantissa)

Where:

  • Characteristic = integer part of log
  • Mantissa = decimal part of log (use table or interpolation)

Example: log N = 2.3010 β†’ Characteristic = 2, Mantissa = 0.3010 β†’ N = 10^2 Γ— 10^0.3010 β‰ˆ 100 Γ— 2 β‰ˆ 200

3. Sample 4-Figure Log Table

This table demonstrates a small part of a 4-figure log table:

Nlog NAntilog
1.000.00001.000
1.010.00431.010
1.020.00861.020
1.030.01281.030
1.040.01701.040
1.050.02121.051
1.060.02531.061
1.070.02941.072
1.080.03341.083
1.090.03741.094
1.100.04141.105

This is **exactly how old 4-figure logarithm tables were constructed**: using linear interpolation or Taylor series for mantissa.

4. Step-by-Step Table Construction

  1. Select numbers N from 1.00 to 9.99 for mantissa table
  2. Normalize to 1 ≀ a < 10
  3. Compute mantissa using log(1 + d) β‰ˆ 0.4343 d for small d
  4. Add characteristic = integer part
  5. Use table for interpolation
  6. Compute antilog as 10^characteristic Γ— 10^mantissa

This method allows you to generate **any log/antilog** table of 4 significant figures programmatically or manually.

5. Proof / Verification

We know by definition: log(N) = log(a Γ— 10^n) = log(a) + log(10^n) = log(a) + n. This is why **tables separate characteristic & mantissa**. Interpolation or series ensures 4-figure precision for log(a). Example verification: log(2.5) - Normalize: 2.5 β†’ a=2.5, n=0 - Mantissa: d = 2.5-1=1.5 β†’ log(2.5) β‰ˆ 0.4343*1.5 = 0.65145 β‰ˆ table value 0.3979 (after refining with series) - Correct characteristic = 0 β†’ log 2.5 β‰ˆ 0.3979

Tip: Always refine mantissa using small corrections if high precision is needed.

6. Real-Life Application

- Engineers and scientists use this formula to create **custom logarithm tables** for calculation in the lab. - Financial analysts can compute **compound interest** without calculators. - Programmers can generate log tables for **embedded systems** where floating point operations are expensive.

Create Your Own 4-Figure Logarithm Table

Create Your Own 4-Figure Logarithm & Antilogarithm Tables

1. Introduction / Utangulizi

Before calculators, mathematicians created logarithm tables manually. To become an expert 4-figure log maker, a student must understand the formulas, steps, and methods. The goal is to compute logarithms and antilogarithms of any number using formulas only.

2. Step 1: Normalize Numbers / Kawaidaisha Nambari

Every number N is expressed as:

N = a Γ— 10^n, where 1 ≀ a < 10 and n ∈ β„€

- a = mantissa base number - n = characteristic (integer part of log) - N = the original number

Example: 250 β†’ 2.5 Γ— 10^2 β†’ characteristic n = 2, mantissa base a = 2.5

3. Step 2: Compute Mantissa / Hesabu ya Mantissa

Use the series expansion to compute log a manually:

log a β‰ˆ 0.4343 Γ— (a - 1) - 0.216 Γ— (a - 1)^2 + 0.115 Γ— (a - 1)^3 - ...

- This formula comes from natural logarithm expansion: ln(1 + x) = x - xΒ²/2 + xΒ³/3 - ... - Multiply by 0.4343 to convert ln β†’ log base 10

Compute log 2.5: a=2.5, d=a-1=1.5 log 2.5 β‰ˆ 0.4343*1.5 - 0.216*1.5Β² + 0.115*1.5Β³ β‰ˆ 0.3979 (mantissa)

4. Step 3: Add Characteristic / Ongeza Nambari Kamili

Combine mantissa and characteristic to get final log:

log N = characteristic + mantissa

log 250 = log(2.5 Γ— 10Β²) = log 2.5 + 2 β‰ˆ 0.3979 + 2 = 2.3979

5. Step 4: Compute Antilog / Hesabu ya Antilog

Given log N = L = characteristic + mantissa, compute:

N = 10^L = 10^characteristic Γ— 10^mantissa

Find N for log N = 2.3979 β†’ characteristic=2, mantissa=0.3979 10Β² Γ— 10^0.3979 β‰ˆ 100 Γ— 2.5 β‰ˆ 250

6. Step 5: Construct Table / Kuunda Jedwali

To build a 4-figure log table manually:

  1. Select numbers N from 1.00 to 9.99 for mantissa table
  2. Normalize each number
  3. Compute mantissa using the series formula
  4. Add characteristic for each power of ten
  5. Compute antilogs if needed using 10^L
  6. Round mantissa/antilog to 4 figures for the table

Sample Table:

NCharacteristicMantissaLog NAntilog N
1.0000.00000.00001.000
1.0100.00430.00431.010
1.0200.00860.00861.020
1.0300.01280.01281.030

7. Multiple Approaches / Njia Nyingi za Hesabu

  • Use series expansion for accurate manual computation
  • Use linear interpolation between known values for intermediate numbers
  • Use geometric progression for rough approximations
  • Check using reverse antilog

8. Proof of Correctness / Uthibitisho

By definition: log(N) = log(a Γ— 10^n) = log(a) + n - Characteristic = n - Mantissa = log(a) from series - Thus table values are consistent with the logarithm definition

9. Tips & Advice / Ushauri

  • Always normalize numbers before computing log
  • Use small increments (0.01 or 0.001) for smoother table
  • Check mantissa with antilog for verification
  • Round off to 4 figures consistently

10. Real-Life Use / Matumizi Halisi

  • Design own calculator or computing device
  • Check computations in engineering or science labs
  • Understand growth rates, decibels, pH calculations
  • Preparation for exam & exam questions without pre-made tables
Composition Practice β€” Set 1

English Composition Practice

Set 1: Questions 1–10. Answer in full sentences. Observe rules for introduction, body, and conclusion.

Set 1: Introduction & Sentence Formation

  1. Question 1: Write an introductory sentence for a composition about "My School".
    Answer: My school is a place of learning where students gain knowledge and skills.
    Rule: Start with a clear subject and general statement. Introduction should grab attention.
  2. Question 2: Begin a paragraph describing your favorite teacher.
    Answer: My favorite teacher is Mr. John because he explains lessons clearly and encourages students.
    Rule: Use simple subject + reason; descriptive adjectives improve clarity.
  3. Question 3: Form a sentence showing your daily school routine.
    Answer: Every morning, I wake up early, eat breakfast, and go to school.
    Rule: Use adverbs of frequency (every morning) + sequence (first, then).
  4. Question 4: Write a complex sentence about a school activity you enjoy.
    Answer: I enjoy playing football during break because it helps me relax and stay healthy.
    Rule: Combine two ideas using conjunctions (because, and) for complex sentences.
  5. Question 5: Create a sentence with a transition word for paragraph flow.
    Answer: Furthermore, the library provides students with many useful books for study.
    Rule: Use linking words (furthermore, moreover, however) to connect ideas.
  6. Question 6: Write an introductory sentence for a narrative composition about a picnic.
    Answer: Last weekend, my family went on a picnic to the countryside, and it was a memorable experience.
    Rule: Specify time (last weekend) and introduce main event.
  7. Question 7: Make a sentence describing a place using adjectives.
    Answer: The park was spacious, green, and full of colorful flowers.
    Rule: Use multiple descriptive adjectives to create vivid imagery.
  8. Question 8: Form a sentence showing cause and effect.
    Answer: Because it rained heavily, the football match was postponed.
    Rule: Use subordinating conjunctions (because, since, as) for cause-effect.
  9. Question 9: Write a sentence using an opinion phrase.
    Answer: In my opinion, students should respect their teachers to learn effectively.
    Rule: Use phrases like "In my opinion", "I believe" to express views.
  10. Question 10: Make a sentence that concludes a paragraph about school life.
    Answer: Overall, school life is enjoyable and teaches important lessons for the future.
    Rule: Use conclusion phrases (Overall, In conclusion) to summarize points.
Logarithm Examples
Example

Calculate the value of

(9304 − 2319) / (0.086 × 41750)

Solution
Number Standard form Logarithm
9304 − 2319 (6.985 × 103) 3.8441
Denominator:
0.086 8.6 × 10−2 1.9345 − 2 = 0.9345
41750 4.1750 × 104 4.6200
Logarithm of numerator 3.8441
Logarithm of denominator 5.5545
Logarithm of quotient 1.8095

Use mathematical tables to read the anti-logarithm of 1.8095, then multiply the answer by 100 (that is, 1).
Therefore, (9304 − 2319) / (0.086 × 41750) = 64.7

Example 4.35

Simplify

(18.1)2 × 0.005678 / √62

Solution
Number Standard form Logarithm
(18.1)2 (3.28 × 102) 2.5166
0.005678 5.678 × 10−3 0.7534
Logarithm of numerator 3.2700
√62 7.8740 0.8963
Logarithm of denominator 0.8963
Logarithm of quotient 2.3737

Use mathematical tables to read the anti-logarithm of 2.3737, then multiply the answer by 100.
Therefore, (18.1)2 × 0.005678 / √62 = 236.5

Reference Book: N/A

Author name: SIR H.A.Mwala Work email: biasharaboraofficials@gmail.com
#MWALA_LEARN Powered by MwalaJS #https://mwalajs.biasharabora.com
#https://educenter.biasharabora.com

:: 1::

β¬… ➑