Advanced Exponential and logarithmic functions

This lesson comprises three (3) master classes focusing on:

• Euler’s number
• Index laws
• Logarithmic laws
• Exponential functions
• Logarithmic functions

Content:

MA-E1.1

• Define logarithms as indices: \( y=a^x \) is equivalent to \( x=\log_{a}y \), and explain why this definition only makes sense when \( a \gt 0 \), \( a \ne 1 \)
• Recognise and sketch the graphs of \( y=ka^x \), \( y=ka^{−x} \) where \( k \) is a constant, and \( y=\log_ax \)
• Recognise and use the inverse relationship between logarithms and exponentials
  • understand and use the fact that \( \log_aa^x=x \) for all real \( x \), and \( a\log_ax=x \) for all \( x \gt 0 \)

ME-E1.2

• Derive the logarithmic laws from the index laws and use the algebraic properties of logarithms to simplify and evaluate logarithmic expressions: \( \log_am + \log_an=\log_a(mn) \), \( \log_am − \log_an=\log_a(\frac{m}{n}) \), \( \log_a(m^n)=n \log_am \), \( \log_aa=1 \), \( \log_a1=0 \), \( \log_a \frac{1}{x}=−\log_ax \)
• Consider different number bases and prove and use the change of base law \( \log_ax=\frac{\log_bx}{\log_ax} \)
• Interpret and use logarithmic scales, for example decibels in acoustics, different seismic scales for earthquake magnitude, octaves in music or pH in chemistry
• 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

ME-F1.3

• Establish and use the formula \( \frac{d(e^x)}{dx}=e^x \)
  • using technology, sketch and explore the gradient function of exponential functions and determine that there is a unique number \( e \approx 2.71828182845 \), for which \( \frac{d(e^x)}{dx}=e^x \) where \( e \) is called Euler’s number
• Apply the differentiation rules to functions involving the exponential function, \( f(x)=ke^{ax} \), where \( k \) and \( a \) are constants
• Work with natural logarithms in a variety of practical and abstract contexts
  • define the natural logarithm \( \ln x=\log_ex \) from the exponential function \( f(x)=e^x \)
  • recognise and use the inverse relationship of the functions \( y=e^x \) and \(y=\ln x \)
  • use the natural logarithm and the relationships \( e\ln x=x \) where \( x \gt 0 \), 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

MA-F1.4

• Solve equations involving indices using logarithms
• Graph an exponential function of the form \( y=a^x \) for \( a \gt 0 \) and its transformations \( y=ka^x+c \) and \( y=ka^{x+b} \) 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
• Establish and use the algebraic properties of exponential functions to simplify and solve problems
• Solve problems involving exponential functions in a variety of practical and abstract contexts, using technology, and algebraically in simple cases
• Graph a logarithmic function \( y=\log_ax \) for \( a \gt 0 \) and its transformations \( y=k\log_ax+c \), using technology or otherwise, where \( k \) and \( c \) are constants
  • recognise that the graphs of \( y=a^x \) and \( y=\log_ax \) are reflections in the line \( y=x \)
• Model situations and solve simple equations involving logarithmic or exponential functions algebraically and graphically
• Identify contexts suitable for modelling by exponential and logarithmic functions and use these functions to solve practical problems