This unit of work addresses the Exponentials and Logarithmic Functions topic from the NSW Stage 6 Year 11 Mathematics Advanced Syllabus (NSW Education Standards Authority [NESA], 2017]). A detailed exploration of the rationale for this unit, prior knowledge and misconceptions alongside the driving pedagogical approaches behind this unit are explained in designated sections of this site.

Below is the relevant syllabus content being addressed throughout the lessons of this unit as well as noted content that would need to be addressed in further lessons. There are 6 lessons in this unit of work as well as a suggested assessment task to be completed at the end of the unit. Additionally, there is a provided vocabulary list and references for all sources used that can be accessed through the headings at the top of the site's page.

A student:

• manipulates and solves expressions using the logarithmic and index laws, and uses logarithms and exponential functions to solve practical problems (MA11-6)

• uses appropriate technology to investigate, organise, model and interpret information in a range of contexts (MA11-8)

• provides reasoning to support conclusions which are appropriate to the context (MA11-9)

(NESA, 2017)

E1.1: Introducing logarithms

Students:

a. define logarithms as indices: y = a^x is equivalent to x = log_a(y), and explain why this definition only makes sense when a > 0, a ≠ 1

b. recognise and sketch the graphs of y = k*a^x, y = k*a^(−x) where k is a constant, and y = log_a(x)

c. recognise and use the inverse relationship between logarithms and exponentials

(i) understand and use the fact that log_a(a)^x = x for all real x, and a^(log_a(x)) = x for all x > 0

E1.2: Logarithmic laws and applications

a. derive the logarithmic laws from the index laws and use the algebraic properties of logarithms to simplify and evaluate logarithmic expressions

• log_a(m) + log_a(n) = log_a(mn), log_a(m) − log_a(n) = log_a(mn), log_a(mn) = n*log_a(m),

• log_a(a) = 1, log_a(1) = 0, log_a(1/x) = −log_a(x)

b. consider different number bases and prove and use the change of base law log_a(x) = log_b(x)/log_a(x)

c. interpret and use logarithmic scales, for example decibels in acoustics, different seismic scales for earthquake magnitude, octaves in music or pH in chemistry (ACMMM154)

d. solve algebraic, graphical and numerical problems involving logarithms in a variety of practical and abstract contexts, including applications from financial, scientific, medical and industrial contexts

E1.3: The exponential function and natural logarithms

a. establish and use the formula d(ex)/dx = e^x (ACMMM100)

• using technology, sketch and explore the gradient function of exponential functions and determine that there is a unique number e ≈ 2.71828182845, for which d(ex)/dx=e^x where e is called Euler’s number

b. apply the differentiation rules to functions involving the exponential function, f(x) = k*e^(a*x), where k and a are constants

c. work with natural logarithms in a variety of practical and abstract contexts

• • define the natural logarithm ln(x) = log_e(x) from the exponential function f(x) = e^x (ACMMM159)

  • recognise and use the inverse relationship of the functions y = e^x and y = ln(x) (ACMMM160)

  • use the natural logarithm and the relationships e^(ln(x)) = x where x > 0, and ln(e^x) = x for all real x in both algebraic and practical contexts

  • use the logarithmic laws to simplify and evaluate natural logarithmic expressions and solve equations

E1.4: Graphs and applications of exponential and logarithmic functions

a. solve equations involving indices using logarithms (ACMMM155)

b. graph an exponential function of the form y = a^x for a > 0 and its transformations y = k*a^x + c and y = k*a^x + b where k, b and c are constants

• • interpret the meaning of the intercepts of an exponential graph and explain the circumstances in which these do not exist

(NESA, 2017)

E1.4: Graphs and applications of exponential and logarithmic functions

c. establish and use the algebraic properties of exponential functions to simplify and solve problems(ACMMM064)

d. solve problems involving exponential functions in a variety of practical and abstract contexts, using technology, and algebraically in simple cases (ACMMM067)

e. graph a logarithmic function y = log_a(x) for a > 0 and its transformations y = k*log_a(x) + c, using technology or otherwise, where k and c are constants

f. recognise that the graphs of y = a^x and y = log_a(x) are reflections in the line y = x

g. model situations and solve simple equations involving logarithmic or exponential functions algebraically and graphically

h. identify contexts suitable for modelling by exponential and logarithmic functions and use these functions to solve practical problems (ACMMM066, ACMMM158)

(NESA, 2017)

1. Lesson 1: Introducing logarithms

2. Lesson 2: Logarithmic laws and applications (Pt. 1)

3. Lesson 3: Logarithmic laws and applications (Pt. 2)

4. Lesson 4: The exponential function and natural logarithms (Pt. 1)

5. Lesson 5: The exponential function and natural logarithms (Pt. 2)

6. Lesson 6: Graphs and applications of exponential and logarithmic functions