#ExponentialLogarithmicFunctions #Year11 #Advanced >[!info]- [Exponential and logarithmic functions | NSW Curriculum Website](https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-advanced-11-12-2024/content/n11/fa576d8dce) >- 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 | Prior knowledge | Used for | | ------------------------------ | ----------------------- | -------------------- | | [[Non-Linear Relationships C]] | - exponential functions | - *repeated content* | | [[Logarithms]] | - log laws | - *repeated content* | ## Exponential functions - Graph the exponential functions $y=k(a^x)$ and $y=k(a^{-x})$ for constants a and k where $a>0$, $a\neq1$ and $k\neq0$, 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 $e\approx2.71828182845...$, 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 $\frac{d}{dx}(a^x)=ka^x$, 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 $a\neq1$, 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 $a\neq1$ - 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 $0<a<1$