Powers and Logarithms

Section A.2 Powers and Logarithms

Subsection 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.

\(\displaystyle e^0=1,\quad q^0=1\)
\(\displaystyle e^{x+y}=e^xe^y, \quad e^{x-y}=\frac{e^x}{e^y}, \quad q^{x+y}=q^xq^y, \quad q^{x-y}=\frac{q^x}{q^y}\)
\(\displaystyle e^{-x}=\frac{1}{e^x}, \quad q^{-x}=\frac{1}{q^x}\)
\(\displaystyle \big(e^x\big)^y=e^{xy}, \quad \big(q^x\big)^y=q^{xy}\)
\(\displaystyle \diff{}{x}e^x=e^x, \quad \diff{}{x}e^{g(x)}=g'(x)e^{g(x)}, \quad \diff{}{x}q^x=(\ln q)\ q^x\)
\(\int e^x\ \dee{x}=e^x+C, \quad \int e^{ax}\ \dee{x}=\frac{1}{a}e^{ax}+C\) 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.

Subsection 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 \(-\infty \lt x \lt \infty\)
\(\displaystyle \log_q x=\frac{\ln x}{\ln q}, \quad \ln x=\frac{\log_p x}{\log_p e}, \quad \log_q x=\frac{\log_p x}{\log_p q}\)
\(\ln 1=0,\quad \ln e=1\)

\(\log_q 1=0,\quad \log_q q=1\)
\(\displaystyle \ln(xy)=\ln x+\ln y, \quad \log_q(xy)=\log_q x+\log_q y\)
\(\displaystyle \ln\big(\frac{x}{y}\big)=\ln x-\ln y, \quad \log_q\big(\frac{x}{y}\big)=\log_q x-\log_q y\)
\(\displaystyle \ln\big(\frac{1}{y}\big)=-\ln y, \quad \log_q\big(\frac{1}{y}\big)=-\log_q y\)
\(\displaystyle \ln(x^y)=y\ln x, \quad \log_q(x^y)=y\log_q x\)
\(\displaystyle \diff{}{x}\ln x = \frac{1}{x}, \quad \diff{}{x}\log_q x = \frac{1}{x\ln q}\)
\(\displaystyle \int \ln x\ \dee{x}= x\ln x-x +C, \quad \int \log_q x\ \dee{x}= x\log_q x-\frac{x}{\ln q} +C\)
\(\lim\limits_{x\rightarrow\infty}\ln x=\infty, \quad \lim\limits_{x\rightarrow0+}\ln x=-\infty\)

\(\lim\limits_{x\rightarrow\infty}\log_q x=\infty, \quad \lim\limits_{x\rightarrow0+}\log_q x=-\infty\)
The graph of \(\log_{10} x\) is given below. The graph of \(\log_q x\text{,}\) for any \(q \gt 1\text{,}\) is similar.

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