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A.2: Poderes y logaritmos

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    118872
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    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.

    expGraph2.svg

    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.

    logGraph10.svg


    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.