SHS1 Additional Mathematics · Semester 1 Week 7: Number and Algebraic Patterns

Lesson notes

Learning Objectives

Indicator: 1.1.1.LI.6 - Establish the relationship between indices and logarithms and use the properties of logarithms to solve related problems in one base.

By the end of the lesson, learners can:

State the definition of a logarithm as the inverse operation of an index (if N = aˣ then logₐ N = x) and convert between index form and logarithmic form in both directions.
State and apply the product rule (logₐ MN = logₐ M + logₐ N), quotient rule (logₐ (M/N) = logₐ M − logₐ N), power rule (logₐ Pⁿ = n logₐ P), and the fact that logₐ a = 1.
Recognise and avoid the common errors in the curriculum, namely that log(M + N) ≠ log M + log N and log(M − N) ≠ log M − log N.
Solve simple indicial equations of the form aˣ = b by taking logarithms of both sides and applying the power rule, working in one base.
Apply logarithmic techniques to solve compound interest and depreciation word problems, including determining the time required for an investment to reach a target amount.

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Curriculum details

Strand: Modelling with Algebra (Strand 1)
Sub-strand: Number and Algebraic Patterns (1.1)
Content standard: 1.1.1.CS.2 - Demonstrate knowledge and understanding of numbers in relation to Surds, Indices and Logarithms. 1.1.1.LO.1 Solve problems involving properties of binary operations. 1.1.1.LO.2 Model and solve real life problems on sets. 1.1.1.LO.3 Expand binomials with positive integral indices and simplify coefficients of the terms. 1.1.1.LO.4 Perform basic operations on surds as well as solve simple indicial and logarithmic equations.
Indicator: 1.1.1.LI.6 - Establish the relationship between indices and logarithms and use the properties of logarithms to solve related problems in one base.
Suggested placement: Semester 1, Week 7 (Week 7 of the year)
Our suggestion, laid out in curriculum order across three terms of twelve weeks. NaCCA does not fix the week, so follow your school's scheme of learning.
Source document note: The official curriculum document has a fault in this entry, so the curriculum text above is transcribed exactly as printed:

content standard text - p.53: the source prints the content-standard code as 1.1.1.CS2 without the separator before 2; the CSV key uses 1.1.1.CS.2 so the otherwise unambiguous second standard can be imported
exemplars - p.61: adjacent duplicated CambriaMath characters were collapsed, but this PDF also maps some equation glyphs to the wrong letter or operator; verify every mathematical expression in this row against the rendered page
Curriculum reference: NaCCA curriculum document, p. 61

Exemplars (from the NaCCA curriculum)

Collaborative learning, Talk for Learning, Building on what others say.
Learning Experience: Learners in groups explore the relationship between indices and logarithms.
Activity 1: Relationship between indices and logarithms. Learners in groups investigate the relationship between indices and logarithms and establish that: If 𝑁 = 𝑎 0 , then log & 𝑁 = 𝑥 log & 𝑎 = 1 log 𝑝 % = 𝑛 log 𝑃 log(𝑁) = log 𝑁 + log 𝑀 B log i C j = log 𝑁 − log 𝑀
Where M, N and P are real numbers.
DEF B NB: (i) ≠ log 𝑁 − log 𝑀 DEF C
log(𝑀 + 𝑁) ≠ log 𝑁 + log 𝑀
Activity 2: Learners in mixed-ability groups solve related problems in one base. Suppose that GHS 5,000 is invested at 6% interest compounded annually. In 𝑡 years, an investment will grow to the amount expressed by the function 𝑆(𝑡) = 5000 ∙ 1.06 G , where 𝑡 is time (in years). How long will it take to accumulate GHS 15,000 in the account?
Expected solutions: You should solve an equation 𝑆(𝑡) = 15000, which is 5000 ∙ 1.06 G = 15000 for unknown 𝑡. Divide both sides of this equation by the initial amount of 5000. You get the equation
1.06 G = 3
Take logarithm base 10 from both sides. You get the equation,
(1.06 G ) = 3
Apply the power rule to the logarithm. You get the equation,
1.06 = 3
" (.,77! Therefore, 𝑡 = = (.()." = 18.8542 (Approximately 19 years) !.(/
Further example: The value 𝑉 of a Range Rover Velar that is 𝑡 years old can be modelled by the function: 𝑉(𝑡) = 25000 ∙ 0.85 G What would the car be worth in 3 years? In how many years will the car be worth GHS 235,000? Activity 3: Learners in mixed-ability groups solve related problems in different bases. What should be one important step when solving logarithm equations with different bases?
Note Change of bases: General rule: Suppose x, a and b are positive numbers with 𝑎, 𝑏 ≠ 1, then,
log & 𝑥 = DEF # & DEF 0 #
Teaching and Learning Resources:
- Textbooks
- Curriculum
- Cardboards
- Reading resource
- Colour pens
- Notebook
- Technological tools
Assessment (1.1.1.AS.6). The document marks these depth-of-knowledge levels for this indicator: Level 4 Extended critical thinking and reasoning.
1. For the rule 𝑦 = 20 × 3 G , a) Complete the table of values. 𝑡 0 1 2 3 𝑦 b) Plot the graph of 𝑦 against 𝑡. c) Find the value 𝑦, correct to 2 decimal places, when: i. 𝑡 = 0.5 𝑡 = 2.5 ii. iii. 𝑡 = 2 2. Suppose that GH₵ 5,000 is invested at 6% iinterest compounded annually. In 𝑡 years an investment will grow to the amount expressed by the function 𝑆(𝑡) = 5000 ∙ 1.06 G where 𝑡 is time (in years). a) Explain how you will use the idea of logarithms to calculate the number of years it will take to double the initial investment b) How long will it take to accumulate GH₵ 15,000 in the account?