General Mathematics JAMB

Indices, Logarithms And Surds

Aperçu

Indices, logarithms, and surds are fundamental concepts in General Mathematics that play a crucial role in various calculations and problem-solving scenarios. Understanding these topics is essential for students to navigate through complex mathematical operations efficiently. This course material will delve deep into the intricacies of indices, logarithms, and surds, providing a comprehensive overview of their principles, applications, and interrelationships.

The primary objective of this course material is to equip students with the necessary skills to apply the laws of indices in calculations effectively. Indices, also known as exponents, govern the way numbers are raised to powers, leading to efficient computations across different numerical scenarios. By mastering the laws of indices, students will be able to simplify complex expressions, manipulate variables with ease, and solve equations involving powers and roots proficiently.

Furthermore, this course material aims to establish a clear relationship between indices and logarithms to enhance students' problem-solving abilities. Logarithms serve as powerful tools that help convert exponential equations into linear form, simplifying calculations and facilitating the solving of intricate mathematical problems. Understanding how logarithms and indices correlate enables students to tackle complex equations, evaluate functions, and analyze growth and decay processes effectively.

In addition to exploring indices and logarithms, this course material will focus on solving problems in different bases using logarithmic functions. Students will learn how to manipulate numbers across various number bases ranging from 2 to 10, understanding the significance of base transformations and their impact on mathematical operations. By mastering logarithmic computations in different bases, students will enhance their numerical fluency and problem-solving skills across diverse mathematical contexts.

Moreover, this course material will delve into the realm of surds, emphasizing the importance of simplifying and rationalizing these irrational numbers. Surds often appear in mathematical expressions involving roots and provide a unique challenge that requires careful manipulation to simplify and integrate seamlessly into calculations. By mastering basic operations on surds, students will develop the skills to simplify square roots, manipulate radical expressions, and solve equations involving irrational numbers efficiently.

Objectifs

  1. Solve Problems In Different Bases In Logarithms
  2. Apply The Laws Of Indices In Calculation
  3. Perform Basic Operations On Surds
  4. Simplify And Rationalize Surds
  5. Establish The Relationship Between Indices And Logarithms In Solving Problems

Note de cours

In this lesson, we will delve into the concepts of Indices, Logarithms, and Surds, which are fundamental topics in General Mathematics. Understanding these topics helps build a strong foundation for more advanced mathematical problems and applications. Each section will cover definitions, laws, and problem-solving techniques to help you master these concepts.

Évaluation de la leçon

Félicitations, vous avez terminé la leçon sur Indices, Logarithms And Surds. Maintenant que vous avez exploré le concepts et idées clés, il est temps de mettre vos connaissances à lépreuve. Cette section propose une variété de pratiques des questions conçues pour renforcer votre compréhension et vous aider à évaluer votre compréhension de la matière.

Vous rencontrerez un mélange de types de questions, y compris des questions à choix multiple, des questions à réponse courte et des questions de rédaction. Chaque question est soigneusement conçue pour évaluer différents aspects de vos connaissances et de vos compétences en pensée critique.

Utilisez cette section d'évaluation comme une occasion de renforcer votre compréhension du sujet et d'identifier les domaines où vous pourriez avoir besoin d'étudier davantage. Ne soyez pas découragé par les défis que vous rencontrez ; considérez-les plutôt comme des opportunités de croissance et d'amélioration.

  1. Simplify the expression \(4^{3} \times 4^{-2}\). A. \(16^{5}\) B. \(16\) C. \(\frac{1}{16}\) D. \(1\) Answer: C. \(\frac{1}{16}\)
  2. Solve for \(x\) in the equation \(2^{x} = 8\). A. \(2\) B. \(3\) C. \(4\) D. \(8\) Answer: B. \(3\)
  3. Evaluate \(\log_{2} 32\). A. \(4\) B. \(5\) C. \(3\) D. \(6\) Answer: B. \(5\)
  4. Simplify \(\sqrt{50}\). A. \(5\sqrt{2}\) B. \(10\sqrt{5}\) C. \(5\sqrt{10}\) D. \(25\) Answer: A. \(5\sqrt{2}\)
  5. If \(\log_{3} y = \frac{1}{2}\), then \(y\) is equal to: A. \(3\) B. \(\frac{3}{2}\) C. \(9\) D. \(\frac{9}{2}\) Answer: C. \(9\)
  6. Simplify \(\sqrt{75} + \sqrt{27}\). A. \(12\) B. \(11\) C. \(10\) D. \(9\) Answer: A. \(12\)
  7. If \(\log_{5} x = 2\), then \(x\) is equal to: A. \(25\) B. \(10\) C. \(125\) D. \(5\) Answer: A. \(25\)
  8. Evaluate \(\frac{2^3 \times 3^2}{2^2 \times 3^3}\). A. \(3\) B. \(\frac{3}{2}\) C. \(2\) D. \(\frac{2}{3}\) Answer: D. \(\frac{2}{3}\)
  9. Simplify \(\sqrt{128} - \sqrt{32}\). A. \(4\sqrt{2}\) B. \(6\sqrt{2}\) C. \(8\sqrt{2}\) D. \(10\sqrt{2}\) Answer: B. \(6\sqrt{2}\)
  10. If \(\log_{7} z = 2\), then \(z\) is equal to: A. \(14\) B. \(21\) C. \(49\) D. \(28\) Answer: C. \(49\)

Livres recommandés

Mathematics for senior secondary schools 1
Mathematics for senior secondary schools 1
Sous-titre: New Edition
Éditeur: Longman Publishers
Année: 2020
ISBN: 978-9788233023
Advanced Mathematics
Advanced Mathematics
Sous-titre: Logarithms and Surds
Éditeur: Pearson Education
Année: 2019
ISBN: 978-9819764214

Questions de révision

Vous vous demandez à quoi ressemblent les questions passées sur ce sujet ? Voici plusieurs questions sur Indices, Logarithms And Surds des années précédentes.

WAEC SSCE - General Mathematics - 2020

Question 1 Rapport


(a) In the diagram, AB is a tangent to the circle with centre O, and COB is a straight line. If CD//AB and < ABE = 40°, find: < ODE. 

 (b) ABCD is a parallelogram in which |\(\overline{CD}\)| = 7 cm, I\(\overline{AD}\)I = 5 cm and < ADC= 125°.

(i) Illustrate the information in a diagram.

(ii) Find, correct to one decimal place, the area of the parallelogram.

(c) If x = \(\frac{1}{2}\)(1 - \(\sqrt{2}\)). Evaluate (2x\(^2\) - 2x).

Détails de la réponse

(a) Finding \( \angle ODE \) from the diagram

Reading the diagram: \(AB\) is a tangent touching the circle at \(A\); \(C\), \(O\) and \(B\) lie on one straight line (so \(CB\) passes through the centre \(O\)); \(E\) is the point where this line \(CB\) meets the circle on the right, so \(CE\) is a diameter. \(CD \parallel AB\) and \( \angle ABE = 40^\circ \).

Step 1: Use the tangent. A radius is perpendicular to a tangent at the point of contact, so \( \angle OAB = 90^\circ \).

In \( \triangle OAB \), \( \angle ABO = 40^\circ \), hence

\[ \angle AOB = 180^\circ - 90^\circ - 40^\circ = 50^\circ. \]

Step 2: Use the parallel chord. The line \(CB\) is a transversal cutting the parallel lines \(AB\) and \(CD\). By alternate angles,

\[ \angle DCB = \angle ABE = 40^\circ, \] so \( \angle DCO = 40^\circ \).

Step 3: Base angles of an isosceles triangle. In \( \triangle OCD \), \(OC = OD\) (both radii), so it is isosceles with

\[ \angle ODC = \angle OCD = 40^\circ. \]

Step 4: Angle in a semicircle. Since \(CE\) is a diameter and \(D\) lies on the circle, the angle it subtends is a right angle:

\[ \angle CDE = 90^\circ. \]

Step 5: Combine. The radius \(OD\) lies inside \( \angle CDE \), so

\[ \angle ODE = \angle CDE - \angle ODC = 90^\circ - 40^\circ = 50^\circ. \]

\( \angle ODE = 50^\circ \).

(b) Parallelogram \(ABCD\)

(i) Illustration. Draw parallelogram \(ABCD\) with vertices labelled in order. Mark side \(DC = 7\ \text{cm}\) along the base and side \(AD = 5\ \text{cm}\) meeting it at \(D\), with the interior angle \( \angle ADC = 125^\circ \) between them. The opposite sides are equal and parallel: \(AB = DC = 7\ \text{cm}\), \(BC = AD = 5\ \text{cm}\), and \( \angle ABC = 125^\circ \), while \( \angle DAB = \angle BCD = 55^\circ \).

(ii) Area. For a parallelogram, area equals the product of two adjacent sides and the sine of the included angle:

\[ \text{Area} = |DC| \times |AD| \times \sin(\angle ADC). \]

\[ \text{Area} = 7 \times 5 \times \sin 125^\circ = 35 \times 0.8192 = 28.67\ \text{cm}^2. \]

Area \( \approx 28.7\ \text{cm}^2 \) (to one decimal place).

(c) Evaluate \( 2x^2 - 2x \) when \( x = \tfrac{1}{2}(1 - \sqrt{2}) \)

First compute \( x^2 \):

\[ x^2 = \left(\frac{1-\sqrt{2}}{2}\right)^2 = \frac{(1-\sqrt{2})^2}{4} = \frac{1 - 2\sqrt{2} + 2}{4} = \frac{3 - 2\sqrt{2}}{4}. \]

Then

\[ 2x^2 = \frac{3 - 2\sqrt{2}}{2}, \qquad 2x = 1 - \sqrt{2} = \frac{2 - 2\sqrt{2}}{2}. \]

Therefore

\[ 2x^2 - 2x = \frac{3 - 2\sqrt{2}}{2} - \frac{2 - 2\sqrt{2}}{2} = \frac{3 - 2\sqrt{2} - 2 + 2\sqrt{2}}{2} = \frac{1}{2}. \]

\( 2x^2 - 2x = \dfrac{1}{2} \).

Voir la réponse
NECO SSCE - General Mathematics - 2005

Question 1 Rapport

Evaluate log 18 + log6 - log16\(^{\frac{1}{2}}\)
Voir la réponse
JAMB UTME - Mathematics - 2018

Question 1 Rapport

Evaluate \(5^{-3}\log^{2} \times 2^{2\log 3}\)

Voir la réponse