Saltar al contenido principal

# A.2: Poderes y logaritmos

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

## A.2.1 Poderes

A continuación,$$x$$ y$$y$$ son números reales arbitrarios,$$q$$ es una constante arbitraria que es estrictamente mayor que cero y$$e$$ es 2.7182818284, a diez decimales.

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

$$\lim\limits_{x\rightarrow\infty}q^x=\infty, \quad \lim\limits_{x\rightarrow-\infty}q^x=0$$si$$q \gt 1$$

$$\lim\limits_{x\rightarrow\infty}q^x=0, \quad \lim\limits_{x\rightarrow-\infty}q^x=\infty$$si$$0 \lt q \lt 1$$

• La gráfica de$$2^x$$ se da a continuación. La gráfica de$$q^x\text{,}$$ para cualquiera$$q \gt 1\text{,}$$ es similar.

## A.2.2 Logaritmos

A continuación,$$x$$ y$$y$$ son números reales arbitrarios que son estrictamente mayores que 0 (salvo que se especifique lo contrario),$$p$$ y$$q$$ son constantes arbitrarias que son estrictamente mayores que uno, y$$e$$ es 2.7182818284, a diez decimales. La notación$$\ln x$$ significa que$$\log_e x\text{.}$$ algunas personas usan$$\log x$$ para significar$$\log_{10} x\text{,}$$ otras lo usan para significar$$\log_e x$$ y otros lo usan para significar$$\log_2 x\text{.}$$

• $$\displaystyle e^{\ln x}=x,\quad q^{\log_q x}=x$$
• $$\ln \big(e^x\big)=x,\quad \log_q \big(q^x\big)=x\quad$$para todos$$-\infty \lt x \lt \infty$$
• $$\displaystyle \log_q x=\frac{\ln x}{\ln q}, \quad \ln x=\frac{\log_p x}{\log_p e}, \quad \log_q x=\frac{\log_p x}{\log_p q}$$
• $$\ln 1=0,\quad \ln e=1$$

$$\log_q 1=0,\quad \log_q q=1$$

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

$$\lim\limits_{x\rightarrow\infty}\log_q x=\infty, \quad \lim\limits_{x\rightarrow0}\log_q x=-\infty$$

• La gráfica de$$\log_{10} x$$ se da a continuación. La gráfica de$$\log_q x\text{,}$$ para cualquiera$$q \gt 1\text{,}$$ es similar.

This page titled A.2: Poderes y logaritmos is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Joel Feldman, Andrew Rechnitzer and Elyse Yeager via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.