2007-02-08T21:38:34Z https://matematicasies.com/ecuaciones-logaritmicas-y-exponenciales-400#comment110 2007-02-08T21:38:34Z

<p><img src='https://matematicasies.com/local/cache-TeX/aceab9169e4b91e1a0601a9d678ac456.png' style="vertical-align:middle;" width="147" height="47" alt="3^{2x} - 5 \cdot 3^x = -6" title="3^{2x} - 5 \cdot 3^x = -6"><br> <br><span class="spip-puce ltr"><b>–</b></span> Realizamos el cambio de variable <img src='https://matematicasies.com/local/cache-TeX/34b637e6e2a99b1b327019f047bd302e.png' style="vertical-align:middle;" width="58" height="42" alt="3^x=t" title="3^x=t"> <br><span class="spip-puce ltr"><b>–</b></span> Entonces <img src='https://matematicasies.com/local/cache-TeX/6d2d9c181a8557fe6734409403243940.png' style="vertical-align:middle;" width="70" height="47" alt="3^{2x} = t^2" title="3^{2x} = t^2"> <br><span class="spip-puce ltr"><b>–</b></span> Sustituimos en la ecuación original <img src='https://matematicasies.com/local/cache-TeX/3c738041285d52a0d836849bb4172029.png' style="vertical-align:middle;" width="110" height="47" alt="t^2 - 5t = -6" title="t^2 - 5t = -6"> <br><span class="spip-puce ltr"><b>–</b></span> Resolvemos la ecuación de segundo grado (con la incógnita t) y obtenemos como soluciones <img src='https://matematicasies.com/local/cache-TeX/b0af76257fd334f5200fa9c2a421688a.png' style="vertical-align:middle;" width="47" height="38" alt="t=2" title="t=2"> y <img src='https://matematicasies.com/local/cache-TeX/b277b7438901594b437aaaca333e415b.png' style="vertical-align:middle;" width="47" height="38" alt="t=3" title="t=3"> <br><span class="spip-puce ltr"><b>–</b></span> Deshacemos el cambio de variable: <img src='https://matematicasies.com/local/cache-TeX/f3ae788e7ab1c31b295eeedc7a8a7911.png' style="vertical-align:middle;" width="58" height="42" alt="3^x = 3" title="3^x = 3"> y <img src='https://matematicasies.com/local/cache-TeX/c70217bb7f356bbd0508bfbbffce38de.png' style="vertical-align:middle;" width="58" height="42" alt="3^x = 2" title="3^x = 2"> y resolvemos las dos ecuaciones resultantes: <br><span class="spip-puce ltr"><b>–</b></span> <img src='https://matematicasies.com/local/cache-TeX/a4174d22a5fc251765278d9b6e01f43a.png' style="vertical-align:middle;" width="147" height="50" alt="3^x = 3 \Longrightarrow \fbox{x=1}" title="3^x = 3 \Longrightarrow \fbox{x=1}"> <br><span class="spip-puce ltr"><b>–</b></span> <img src='https://matematicasies.com/local/cache-TeX/c70217bb7f356bbd0508bfbbffce38de.png' style="vertical-align:middle;" width="58" height="42" alt="3^x = 2" title="3^x = 2"> (como 2 no se puede expresar como potencia de 3, se resuelve tomando logaritmos) <br><span class="spip-puce ltr"><b>–</b></span> <img src='https://matematicasies.com/local/cache-TeX/77ca0ea1a1c8abb8d6201f94b851e785.png' style="vertical-align:middle;" width="505" height="67" alt="3^x = 2 \Longrightarrow \log{3^x}=\log{2} \Longrightarrow x \cdot \log{3} = \log{2} \Longrightarrow {\bf x=\frac{log2}{log3}}" title="3^x = 2 \Longrightarrow \log{3^x}=\log{2} \Longrightarrow x \cdot \log{3} = \log{2} \Longrightarrow {\bf x=\frac{log2}{log3}}"></p>