
06/08/2025
📘 Course Title: General Mathematics
📎 Course Code: MTH 111
🏫 Programme: National Diploma (ND) in Public Health Technology
⏱️ Duration: 30 Hours (Lecture Only)
🎓 Credit Unit: 2 Units
🔷 Lecture 2: Logarithms | Darasi na 2: Logaritms
🎯 Objectives | Manufar Darasi:
🔹 To understand the concept and laws of logarithms.
Domin fahimtar ma’anar logaritm da dokokinta.
🔹 To convert between exponential and logarithmic forms.
Domin sauya matsayi tsakanin tsarin ƙarfi (exponential) da tsarin logaritm.
🔹 To apply logarithms in simplifying complex calculations in public health.
Domin amfani da logaritm wajen sauƙaƙe lissafi a fannin kiwon lafiya.
📘 Content | Abun ciki:
📌 1. What is a Logarithm? | Menene Logaritm?
🔹 A logarithm answers the question: To what power must a number (base) be raised to get another number?
Logaritm yana amsa tambayar: Sau nawa ne za a ɗaga lamba (tushe) don a samu wata lamba?
🔸 Example: log₁₀(100) = 2 because 10² = 100
🔹 Misali: log₁₀(100) = 2 domin 10² = 100
📌 2. Converting Between Exponential and Logarithmic Form | Sauya Daga Tsari zuwa Wani:
🔹 Exponential form: a^x = N
Tsarin ƙarfi: a^x = N
🔹 Logarithmic form: logₐ(N) = x
Tsarin logaritm: logₐ(N) = x
🔸 Example: 2³ = 8 → log₂(8) = 3
🔹 Misali: 2³ = 8 → log₂(8) = 3
📌 3. Laws of Logarithms | Dokokin Logaritm:
✅ a. Product Rule (Ƙa’idar Ƙari):
🔹 logₐ(xy) = logₐ(x) + logₐ(y)
🔸 Example: log₁₀(100 × 10) = log₁₀(100) + log₁₀(10) = 2 + 1 = 3
🔹 Misali: log₁₀(100 × 10) = log₁₀(100) + log₁₀(10) = 2 + 1 = 3
✅ b. Quotient Rule (Ƙa’idar Rabo):
🔹 logₐ(x/y) = logₐ(x) - logₐ(y)
🔸 Example: log₁₀(100/10) = log₁₀(100) - log₁₀(10) = 2 - 1 = 1
🔹 Misali: log₁₀(100/10) = log₁₀(100) - log₁₀(10) = 2 - 1 = 1
✅ c. Power Rule (Ƙa’idar Ƙarfi):
🔹 logₐ(xⁿ) = n × logₐ(x)
🔸 Example: log₁₀(10³) = 3 × log₁₀(10) = 3 × 1 = 3
🔹 Misali: log₁₀(10³) = 3 × log₁₀(10) = 3 × 1 = 3
✅ d. Change of Base Formula (Sauya Tushe):
🔹 logᵦ(N) = logₐ(N) / logₐ(ᵦ)
🔸 Example: log₂(8) = log₁₀(8) / log₁₀(2) = 0.903 / 0.301 = 3
🔹 Misali: log₂(8) = log₁₀(8) / log₁₀(2) = 0.903 / 0.301 = 3
🩺 Applications in Public Health | Amfani a Fannin Lafiya:
✅ 1. pH Scale (Acidity/Alkalinity):
🔹 pH = -log₁₀[H⁺]
🔹 pH = -log₁₀[H⁺]
🔸 If H⁺ = 0.001 → pH = -log(0.001) = 3
🔹 Idan H⁺ = 0.001 → pH = -log(0.001) = 3
🔹 pH yana bayyana tsananin ɗaci ko ɗanɗano a cikin ruwa ko jini.
Yana da amfani wajen tantance sinadarai masu illa ga lafiyar jiki.
✅ 2. Population Growth (Ƙarar Yawan Jama'a):
🔹 N = N₀ × eʳᵗ → log(N) = log(N₀) + rt
🔹 Yawan jama'a = yawan farko × eʳᵗ → log(N) = log(N₀) + rt
🔸 Example: If a town has 1000 people and grows at 5% per year, after 3 years:
Misali: Idan gari na da mutane 1000, yana ƙaruwa da 5% a shekara, bayan shekaru 3:
🔹 N = 1000 × e^(0.05 × 3) = 1000 × e⁰.¹⁵ ≈ 1161.83
🔹 N = 1000 × e⁰.¹⁵ ≈ 1161.83
🧠 Practice | Aiki da Hannu:
🔹 Students use log tables or calculators to:
🔸 Simplify complex multiplications.
🔸 Solve exponential/logarithmic equations.
🔸 Calculate pH and estimate growth.
🔹 Dalibai su yi amfani da jadawalin log ko kalkuleta su:
🔸 Sauƙaƙe ninkuwa masu wuya.
🔸 Warware lissafi na exponential da logaritm.
🔸 Lissafa pH da hasashen yawan jama'a.
🧰 Tools Needed | Kayan Aiki:
✅ Textbooks | Litattafai
✅ Scientific calculators | Kalkuleta mai fasaha
✅ Whiteboard & markers | Allon rubutu da alƙalami
✅ Log tables | Jadawalin Logaritm
✅ Evaluation | Kwatance:
📝 Classwork (Lissafi a aji)
📚 Assignment (Aikin gida)
🧪 Test (Jarabawa)
Lecture note by Idris M Muhammad ✍️
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