Exponential and Logarithmic Functions

exponentiallogarithmicfunctions year11 advanced

Exponential and logarithmic functions | NSW Curriculum Website

• MAV-11-07 applies exponential and logarithmic laws to simplify expressions, solve equations and prove results
• MAV-11-08 analyses graphs of exponential and logarithmic functions

📖 Prior Knowledge

Content	Prerequisite relationships
Financial Mathematics B	- Apply the compound interest formula → Solve exponential and logarithmic equations
Indices A	- Apply index laws to algebraic terms → Define a logarithm
Non-linear Relationships A	- Identify parabolas and exponential curves → Graph exponentials and identify their features
Non-linear Relationships B	- Graph exponentials and identify their features (revisited content)
Indices B	- Convert between negative and positive indices → Translate between index and logarithmic form
Indices C	- Describe and use fractional indices → Translate between index and logarithmic form
Equations C	- Solve quadratic equations by factorisation → Solve exp/log equations reducible to quadratics
Non-Linear Relationships C	- Graph exponentials and describe transformations (revisited content)
Logarithms	- Define a logarithm (revisited content) - Translate between index and logarithmic form (revisited content) - Compare exponential and logarithmic graphs (revisited content) - Apply the laws of logarithms (revisited content) - Use standard logarithmic results (revisited content) - Evaluate and simplify logarithmic expressions (revisited content) - Solve exponential and logarithmic equations (revisited content) - Use the inverse log identities (revisited content)
Introduction to Differentiation	- Define the derivative → Identify Euler’s number e - Apply the power rule → Recognise $e^x$ is its own derivative

Exponential functions

• Graph the exponential functions $y=k(a^x)$ and $y=k(a^{-x})$ for constants a and k where $a>0$, $a\ne 1$ and $k\ne 0$, and identify its asymptote, y-intercept, domain and range
• Describe the behaviour of $y=k(a^x)$ and $y=k(a^{-x})$ as $x\to \infty$ and $x\to -\infty$
• Examine the gradient of the tangent to the curve $y=a^x$ at its $y$-intercept for varying values of a, and verify using graphing applications that there is a unique number $e\approx \mathrm{2.71828182845...}$, such that the gradient of the tangent to $y=e^x$ at $x=0$ is 1, and call this number e Euler’s number
• Examine the gradient function of $y=a^x$ with graphing applications and identify that the gradient function is $\frac{d}{dx}(a^x)=k{a}^x$, for some constant k depending only on $a$
• Conclude that Euler’s number, e, is a unique number such that $\frac{d}{dx}(e^x)=e^x$, that is, $e^x$ is its own derivative

Logarithmic functions

• Define the logarithm of a number $y$, where $y>0$, to any positive base $a$ as the index to which $a$ is raised to give $y$
• Use the notation ${\log }_{a}y$ for the logarithm of $y$ to the base $a$
• Define the natural logarithm $\ln a={\log }_{e}a$
• Use digital tools to determine rational and irrational values of exponential and logarithmic expressions
• Explain that $y=a^x$ is equivalent to $x={\log }_{a}y$ for $a>0$ and $a\ne 1$, and use the equivalence to solve equations of the form $a^x=b$, for $a>0$
• Recognise and use the logarithmic properties: ${\log }_a{a}^x=x$ for all real x, $a^{{\log }_{a}x}=x$ where $x>0$
• Derive the laws of logarithms from the laws of indices ${\log }_{a}m+{\log }_{a}n={\log }_a(mn)$, ${\log }_{a}m-{\log }_{a}n={\log }_a(\frac{m}{n})$ and ${\log }_a(m^n)=n{\log }_{a}m$
• Justify the logarithmic results ${\log }_{a}a=1$, ${\log }_{a}1=0$ and ${\log }_a(\frac{1}{x})=-{\log }_{a}x$
• Apply the logarithmic laws and results to simplify expressions, solve equations and prove results, using digital tools where necessary
• Prove the change of base rule ${\log }_{a}x=\frac{{\log }_{b}x}{{\log }_{b}a}$ and use it to solve problems
• Graph the logarithmic function $y={\log }_{a}x$ for $a>0$ and $a\ne 1$
• Use graphing applications to identify the graphs of $y=a^x$ and $y={\log }_{a}x$ as reflections of each other in the line $y=x$, in cases where $a>1$ and $0<a<1$