From 1de15dd25eb855fedb0bcf140aaf87f293ff902d Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Sat, 16 Mar 2019 18:01:35 +1100 Subject: [PATCH] [methods] formatting adjustments for exp/log notes --- methods/stuff.md | 39 +++++++++++++++++++++++---------------- methods/stuff.pdf | Bin 91321 -> 142203 bytes 2 files changed, 23 insertions(+), 16 deletions(-) diff --git a/methods/stuff.md b/methods/stuff.md index bf8fbb0..78c9897 100644 --- a/methods/stuff.md +++ b/methods/stuff.md @@ -12,15 +12,17 @@ header-includes: \pagenumbering{gobble} ## Index laws -$a^m \times a^n = a^{m+n}$ -$a^m \div a^n = a^{m-n}4$ -$(a^m)^n = a^{_mn}$ -$(ab)^m = a^m b^m$ -${({a \over b})}^m = {a^m \over b^m}$ +\begin{equation}\begin{split} + a^m \times a^n & = a^{m+n} \\ + a^m \div a^n & = a^{m-n}4 \\ + (a^m)^n & = a^{_mn} \\ + (ab)^m & = a^m b^m \\ + {({a \over b})}^m & = {a^m \over b^m} +\end{split}\end{equation} ## Fractional indices -$^n\sqrt{x}=x^{1/n}$ +$$^n\sqrt{x}=x^{1/n}$$ ## Logarithms @@ -44,16 +46,20 @@ $$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$ ## Logarithm laws -$\log_a(mn) = \log_am + \log_an$ -$\log_a({m \over n}) = \log_am - \log_an$ -$\log_a(m^p) = p\log_am$ -$\log_a(m^{-1}) = -\log_am$ -$\log_a1 = 0$ and $\log_aa = 1$ +\begin{equation}\begin{split} + \log_a(mn) & = \log_am + \log_an \\ + \log_a({m \over n}) & = \log_am - \log_a \\ + \log_a(m^p) & = p\log_am \\ + \log_a(m^{-1}) & = -\log_am \\ + \log_a1 = 0 & \text{ and } \log_aa = 1 +\end{split}\end{equation} ## Inverse functions -Inverse of $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x$ is $f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=log_ax$ +For $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x$, inverse is: + +$$f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=log_ax$$ ## Euler's number @@ -76,9 +82,9 @@ For continuous decay, $k < 0$ ## Graphing expomnential functions -$$f(x)=Aa^{k(x-b)} + c, \quad \vert a > 1$$ +$$f(x)=Aa^{k(x-b)} + c, \quad \vert \> a > 1$$ -- **$y$-intercept** at $(0, {{1+c} \over {a^b}})$ +- **$y$-intercept** at $(0, A \cdot a^{-kb}+c)$ - **horizontal asymptote** at $y=c$ - **domain** is $\mathbb{R}$ - **range** is $(c, \infty)$ @@ -92,10 +98,11 @@ $log_e x$ is the inverse of $e^x$ (reflection across $y=x$) $$f(x)=A \log_a k(x-b) + c$$ where + - **domain** is $(b, \infty)$ - **range** is $\mathbb{R}^+$ - **vertical asymptote** at $x=b$ - $y$-intercept exists if $b<0$ -- dilation of factor $A$ from $x$-axis (reflection across $x$-axis when $A < 0$) -- dilation of factor $1 \over k$ from $y$-axis (reflection across $y$-axis when $k < 0$) +- dilation of factor $A$ from $x$-axis +- dilation of factor $1 \over k$ from $y$-axis diff --git a/methods/stuff.pdf b/methods/stuff.pdf index c62aeafef1b4043a3279b03c93e7fd3a92816cbf..a15caf88d53e6688361b98525c5c670f69250ea0 100644 GIT binary patch delta 138898 zcmV)HK)t`Y$p!n?2qz^_L`E$!E;R}z(A34)3N<$%Fd%PYY6?6&3NK7$ZfA68ATuyD zIFs>kD1X&jS#ujV5`NdO7(az7hgjT4t}3T0-byM}Tbp&B;yjF~k;FC0u}R97^6R$& 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