\section{Exponentials \& Logarithms}

\subsubsection*{Logarithmic identities}

\begin{align*}
  \log_b (xy) &= \log_b x + \log_b y \\
  \log_b x^n &= n \log_b x \\
  \log_b y^{x^n} &= x^n \log_b y \\
  \log_a(\frac{m}{n}) &= \log_am - \log_a \\
  \log_a(m^{-1}) & = -\log_am \\
  \log_b c &= \frac{\log_a c}{\log_a b}
\end{align*}

\subsubsection*{Index identities}

\begin{align*}
  b^{m+n} &= b^m \cdot b^n \\
  (b^m)^n &= b^{m \cdot n} \\
  (b \cdot c)^n &= b^n \cdot c^n \\
  {b^m \div a^n} &= {b^{m-n}}
\end{align*}

\subsection*{Inverse functions}

For \(f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x\), inverse is:

\[f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=\log_ax\]

\subsection*{Euler's number \(e\)}

\[e= \lim_{n \rightarrow \infty} \left(1 + \dfrac{1}{n}\right)^n\]

\subsection*{Modelling}

\[A = A_0 e^{kt}\]

\begin{itemize}
\tightlist
\item
  \(A_0\) is initial value
\item
  \(t\) is time taken
\item
  \(k\) is a constant
\item
  For continuous growth, \(k > 0\)
\item
  For continuous decay, \(k < 0\)
\end{itemize}

\subsection*{Graphing exponential functions}

\[f(x)=Aa^{k(x-b)} + c, \quad \vert \> a > 1\]

\begin{itemize}
\tightlist
\item
  \textbf{\(y\)-intercept} at \((0, A \cdot a^{-kb}+c)\) as
  \(x \rightarrow \infty\)
\item
  \textbf{horizontal asymptote} at \(y=c\)
\item
  \textbf{domain} is \(\mathbb{R}\)
\item
  \textbf{range} is \((c, \infty)\)
\item
  dilation of factor \(|A|\) from \(x\)-axis
\item
  dilation of factor \(1 \over k\) from \(y\)-axis
\end{itemize}

\begin{tikzpicture}
  \begin{axis}[restrict x to domain=-0.9:0.9, axis y line = middle, yticklabels={,,}, xticklabels={,,}, enlargelimits, ticks=none]
    \addplot[red, thick, smooth, samples=100]  plot (\x, {pow(2,x)}) node[below, pos=1] {\(2^x\)};
    \addplot[blue, thick, smooth, samples=100] plot (\x, {pow(3,x)}) node[left, pos=1] {\(3^x\)};
    \addplot[orange, thick, smooth, samples=100] plot (\x, {pow(e,x)}) node[below, pos=1] {\(e^x\)};
    \addplot[mark=*] coordinates {(0,1)} node[above left]{\((0,1)\)} ;
    \addplot[purple, ultra thick, dashed] plot (\x, 0) node[black, below, font=\footnotesize, pos=0.75] {\(y=0\)};
  \end{axis}
\end{tikzpicture}

\subsection*{Graphing logarithmic functions}

\(\log_e x\) is the inverse of \(e^x\) (reflection across \(y=x\))

\[f(x)=A \log_a k(x-b) + c\]

where

\begin{itemize}
\tightlist
\item
  \textbf{domain} is \((b, \infty)\)
\item
  \textbf{range} is \(\mathbb{R}\)
\item
  \textbf{vertical asymptote} at \(x=b\)
\item
  \(y\)-intercept exists if \(b<0\)
\item
  dilation of factor \(|A|\) from \(x\)-axis
\item
  dilation of factor \(1 \over k\) from \(y\)-axis
\end{itemize}
\begin{tikzpicture}
  \begin{axis}[axis lines=middle, xmin=-0.5, xmax=5, ymin=-2, ymax=3, ticks=none]
    \addplot[purple, ultra thick, dashed] coordinates {(0,-1.8) (0,2.8)} node[black, below right, pos=0.75, font=\footnotesize] {\(x=0\)};
    \addplot[orange,thick,domain=0.01:4,smooth,samples=100] {ln(x)} node[right, pos=1] {\(\log_e x\)};
    \addplot[red,thick,domain=0.01:4,smooth,samples=100] {log2(x)} node[right, pos=1] {\(\log_2 x\)};
    \addplot[blue,thick,domain=0.01:4,smooth,samples=100] {ln(x)/ln(3)} node[below right, pos=1] {\(\log_3 x\)};
    \addplot[mark=*] coordinates {(1,0)} node[above left]{\((0,1)\)} ;
  \end{axis}
\end{tikzpicture}

\subsection*{Finding equations}

\colorbox{cas}{On CAS:}
\includegraphics[width=0.78125in]{graphics/cas-simultaneous.png}
