A.2: Powers and Logarithms

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A.2.1 Powers

In the following, $x$ and $y$ are arbitrary real numbers, $q$ is an arbitrary constant that is strictly bigger than zero and $e$ is 2.7182818284, to ten decimal places.

$\displaystyle e^0=1,\quad q^0=1$
$\displaystyle e^{x+y}=e^xe^y, \quad e^{x-y}=\frac{e^x}{e^y}, \quad q^{x+y}=q^xq^y, \quad q^{x-y}=\frac{q^x}{q^y}$
$\displaystyle e^{-x}=\frac{1}{e^x}, \quad q^{-x}=\frac{1}{q^x}$
$\displaystyle \big(e^x\big)^y=e^{xy}, \quad \big(q^x\big)^y=q^{xy}$
$\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}e^x=e^x, \quad \frac{\mathrm{d}}{\mathrm{d}x}e^{g(x)}=g'(x)e^{g(x)}, \quad \frac{\mathrm{d}}{\mathrm{d}x}q^x=(\ln q)\ q^x$
$\int e^x\ \mathrm{d}{x} =e^x+C, \quad \int e^{ax}\ \mathrm{d}{x} =\frac{1}{a}e^{ax}+C$ if $a\ne 0$
$\displaystyle e^x =\sum\limits_{n=0}^\infty\frac{x^n}{n!}$
$\lim\limits_{x\rightarrow\infty}e^x=\infty, \quad \lim\limits_{x\rightarrow-\infty}e^x=0$
$\lim\limits_{x\rightarrow\infty}q^x=\infty, \quad \lim\limits_{x\rightarrow-\infty}q^x=0$ if $q \gt 1$

$\lim\limits_{x\rightarrow\infty}q^x=0, \quad \lim\limits_{x\rightarrow-\infty}q^x=\infty$ if $0 \lt q \lt 1$
The graph of $2^x$ is given below. The graph of $q^x\text{,}$ for any $q \gt 1\text{,}$ is similar.

$expGraph2.svg$

A.2.2 Logarithms

In the following, $x$ and $y$ are arbitrary real numbers that are strictly bigger than 0 (except where otherwise specified), $p$ and $q$ are arbitrary constants that are strictly bigger than one, and $e$ is 2.7182818284, to ten decimal places. The notation $\ln x$ means $\log_e x\text{.}$ Some people use $\log x$ to mean $\log_{10} x\text{,}$ others use it to mean $\log_e x$ and still others use it to mean $\log_2 x\text{.}$

$\displaystyle e^{\ln x}=x,\quad q^{\log_q x}=x$
$\ln \big(e^x\big)=x,\quad \log_q \big(q^x\big)=x\quad$ for all $-\infty \lt x \lt \infty$
$\displaystyle \log_q x=\frac{\ln x}{\ln q}, \quad \ln x=\frac{\log_p x}{\log_p e}, \quad \log_q x=\frac{\log_p x}{\log_p q}$
$\ln 1=0,\quad \ln e=1$
$\log_q 1=0,\quad \log_q q=1$
$\displaystyle \ln(xy)=\ln x+\ln y, \quad \log_q(xy)=\log_q x+\log_q y$
$\displaystyle \ln\big(\frac{x}{y}\big)=\ln x-\ln y, \quad \log_q\big(\frac{x}{y}\big)=\log_q x-\log_q y$
$\displaystyle \ln\big(\frac{1}{y}\big)=-\ln y, \quad \log_q\big(\frac{1}{y}\big)=-\log_q y$
$\displaystyle \ln(x^y)=y\ln x, \quad \log_q(x^y)=y\log_q x$
$\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\ln x = \frac{1}{x}, \quad \frac{\mathrm{d}}{\mathrm{d}x}\log_q x = \frac{1}{x\ln q}$
$\displaystyle \int \ln x\ \mathrm{d}{x} = x\ln x-x +C, \quad \int \log_q x\ \mathrm{d}{x} = x\log_q x-\frac{x}{\ln q} +C$
$\lim\limits_{x\rightarrow\infty}\ln x=\infty, \quad \lim\limits_{x\rightarrow0}\ln x=-\infty$
$\lim\limits_{x\rightarrow\infty}\log_q x=\infty, \quad \lim\limits_{x\rightarrow0}\log_q x=-\infty$
The graph of $\log_{10} x$ is given below. The graph of $\log_q x\text{,}$ for any $q \gt 1\text{,}$ is similar.

$logGraph10.svg$