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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

Solve Problems In Different Bases In Logarithms
Apply The Laws Of Indices In Calculation
Perform Basic Operations On Surds
Simplify And Rationalize Surds
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.

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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.

Simplify the expression $4^{3} \times 4^{-2}$. A. $16^{5}$ B. $16$ C. $\frac{1}{16}$ D. $1$ Answer: C. $\frac{1}{16}$
Solve for $x$ in the equation $2^{x} = 8$. A. $2$ B. $3$ C. $4$ D. $8$ Answer: B. $3$
Evaluate $\log_{2} 32$. A. $4$ B. $5$ C. $3$ D. $6$ Answer: B. $5$
Simplify $\sqrt{50}$. A. $5\sqrt{2}$ B. $10\sqrt{5}$ C. $5\sqrt{10}$ D. $25$ Answer: A. $5\sqrt{2}$
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. $9$
Simplify $\sqrt{75} + \sqrt{27}$. A. $12$ B. $11$ C. $10$ D. $9$ Answer: A. $12$
If $\log_{5} x = 2$, then $x$ is equal to: A. $25$ B. $10$ C. $125$ D. $5$ Answer: A. $25$
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. $\frac{2}{3}$
Simplify $\sqrt{128} - \sqrt{32}$. A. $4\sqrt{2}$ B. $6\sqrt{2}$ C. $8\sqrt{2}$ D. $10\sqrt{2}$ Answer: B. $6\sqrt{2}$
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 Sous-titre New Edition Éditeur Longman Publishers Année 2020 ISBN 978-9788233023
	Advanced Mathematics Sous-titre Logarithms and Surds Éditeur Pearson Education Année 2019 ISBN 978-9819764214

Questions précédentes

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

JAMB UTME - Mathematics - 2018 (Cliquez pour passer l'évaluation complète)

Question 1 Rapport

Evaluate 5 $^{- 3}$ log $^{2}$ $\times$ 2 $^{2 \log 3}$

\frac{1}{8}

\frac{1}{8}

\frac{2}{5}

Voir la réponse

NECO SSCE - General Mathematics - 2005 (Cliquez pour passer l'évaluation complète)

Question 1 Rapport

Evaluate log 18 + log⁶ - log16$^{\frac{1}{2}}$

log 3

log 9

log 20

3 log 2

3 log 3

Voir la réponse

WAEC SSCE - General Mathematics - 2020 (Cliquez pour passer l'évaluation complète)

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.

Réponse

(a)

<ABE = <OBC (tangent and radius form a right angle)
<ABE = 40° (given)
Therefore, <OBC = 40°.
Since CD//AB, then <CDE = <ABE = 40° (alternate angles)
<ODE = <OBC + <CDE = 40° + 40° = 80°.

(b)

(i)

  A ----------- B
   \           / 
    \         /  
     \       /   
      \     /    
       \   /     
        \ /      
        D-------C
             (125°)

(ii)

The area of the parallelogram is given by A = base x height. Since AD || BC, then the height of the parallelogram is given by the perpendicular distance between AD and BC, which is 7 cm. To find the length of the base, we use the cosine rule:

|$\overline{AD}$|² = |$\overline{AB}$|² + |$\overline{BD}$|² - 2|$\overline{AB}$||$\overline{BD}$| cos(ADC)
5² = x² + (x+7)² - 2x(x+7)cos(125°)
25 = 2x² + 14x + 49 + 2x(x+7)(-0.5736)
25 = 2x² + 14x + 49 - 1.1472x² - 10.0317x - 20.7032
0 = 0.8528x² + 4.0317x + 4.7032

Using the quadratic formula: x = (-b ± ?(b² - 4ac))/2a, we have:
x = (-4.0317 ± ?(4.0317² - 4(0.8528)(4.7032)))/(2(0.8528))
x = (-4.0317 ± 4.9988)/1.7056
x = 0.6559 or x = -3.6582

Since x represents a length, we discard the negative solution. Therefore, x = 0.6559 cm.

So, the area of the parallelogram is A = base x height = 0.6559 cm x 7 cm = 4.5913 cm² (to one decimal place).

(c)

x = $\frac{1}{2}$(1 - $\sqrt{2}$)

2x² - 2x = 2$\left(\frac{1}{2}$(1 - $\sqrt{2}$)²\) - 2(1 - $\sqrt{2}$)
= (1 - 2$\sqrt{2}$ + 2) - 2 + 2$\sqrt{2}$
= 1 + 2$\sqrt{2}$.

Therefore, 2x² - 2x = 1 + 2$\sqrt{2}$.

Détails de la réponse

(a)

(b)

(i)

  A ----------- B
   \           / 
    \         /  
     \       /   
      \     /    
       \   /     
        \ /      
        D-------C
             (125°)

(ii)

Using the quadratic formula: x = (-b ± ?(b² - 4ac))/2a, we have:
x = (-4.0317 ± ?(4.0317² - 4(0.8528)(4.7032)))/(2(0.8528))
x = (-4.0317 ± 4.9988)/1.7056
x = 0.6559 or x = -3.6582

Since x represents a length, we discard the negative solution. Therefore, x = 0.6559 cm.

So, the area of the parallelogram is A = base x height = 0.6559 cm x 7 cm = 4.5913 cm² (to one decimal place).

(c)

x = $\frac{1}{2}$(1 - $\sqrt{2}$)

2x² - 2x = 2$\left(\frac{1}{2}$(1 - $\sqrt{2}$)²\) - 2(1 - $\sqrt{2}$)
= (1 - 2$\sqrt{2}$ + 2) - 2 + 2$\sqrt{2}$
= 1 + 2$\sqrt{2}$.

Therefore, 2x² - 2x = 1 + 2$\sqrt{2}$.

Voir la réponse