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