2007-02-15T00:43:53Z https://www.matematicasies.com/ecuaciones-logaritmicas-y,816#comment117 2007-02-15T00:43:53Z

<p> </p><p class="spip" style="text-align:center;"><img src='https://www.matematicasies.com/local/cache-TeX/3c9b0190de685417031c8877a978782c.png' style="vertical-align:middle;" width="202" height="47" alt="7^{2x+3} - 8 \cdot 7^{x+1} + 1 = 0" title="7^{2x+3} - 8 \cdot 7^{x+1} + 1 = 0"></p> <p>Hacemos el cambio de variable: <img src='https://www.matematicasies.com/local/cache-TeX/9f0e46d384ecdfec53eaf8ad79d5b264.png' style="vertical-align:middle;" width="58" height="42" alt="7^x = t" title="7^x = t"><br> <img src='https://www.matematicasies.com/local/cache-TeX/3d1d9848495d693d7f6ebc44b5319ce8.png' style="vertical-align:middle;" width="373" height="47" alt="7^{2x+3} = 7^{2x} \cdot 7^3 = (7^x)^2 \cdot 7^3 = t^2 \cdot 7^3 = 343t^2" title="7^{2x+3} = 7^{2x} \cdot 7^3 = (7^x)^2 \cdot 7^3 = t^2 \cdot 7^3 = 343t^2"><br> <img src='https://www.matematicasies.com/local/cache-TeX/87d3a9fbbdbfdd48ecb8719ee2f817cf.png' style="vertical-align:middle;" width="173" height="47" alt="7^{x+1} = 7^x + 7^1 = 7t" title="7^{x+1} = 7^x + 7^1 = 7t"><br> Sustituimos en la ecuación original:<br class="autobr"> </p><p class="spip" style="text-align:center;"><img src='https://www.matematicasies.com/local/cache-TeX/907ea05f23230ba85b95e0a64eb325b3.png' style="vertical-align:middle;" width="183" height="47" alt="343t^2 - 8 \cdot 7t + 1 = 0" title="343t^2 - 8 \cdot 7t + 1 = 0"></p> <br class="autobr"> Resolvemos la ecuación de segundo grado (incógnita t)<br> <p class="spip" style="text-align:center;"><img src='https://www.matematicasies.com/local/cache-TeX/f6fdc87489b9bb5509b5f25e493c2f9f.png' style="vertical-align:middle;" width="168" height="47" alt="343t^2 - 56t + 1 = 0" title="343t^2 - 56t + 1 = 0"></p> <br class="autobr"> <p class="spip" style="text-align:center;"><img src='https://www.matematicasies.com/local/cache-TeX/8be96cf92b87266b6dd211b687e9572a.png' style="vertical-align:middle;" width="322" height="75" alt="t= \frac{56 \pm \sqrt{56^2 - 4 \cdot 343 \cdot 1}}{2 \cdot 343} = \frac{56 \pm 42}{686}" title="t= \frac{56 \pm \sqrt{56^2 - 4 \cdot 343 \cdot 1}}{2 \cdot 343} = \frac{56 \pm 42}{686}"></p> <br class="autobr"> <p class="spip" style="text-align:center;"><img src='https://www.matematicasies.com/local/cache-TeX/070d7914d98173a5a56ecb673320123e.png' style="vertical-align:middle;" width="143" height="65" alt="\left[t=\frac{1}{7} \:, \: t=\frac{1}{49} \right]" title="\left[t=\frac{1}{7} \:, \: t=\frac{1}{49} \right]"></p> <br class="autobr"> Deshacemos el cambio de variable y obtenemos dos ecuaciones: <br> <p><span class="spip-puce ltr"><b>–</b></span> <img src='https://www.matematicasies.com/local/cache-TeX/0ed3f9724868a33b38f7277fe635fdd8.png' style="vertical-align:middle;" width="275" height="65" alt="7^x = \frac{1}{7} \longrightarrow 7^x = 7^{-1} \longrightarrow \fbox{x=-1}" title="7^x = \frac{1}{7} \longrightarrow 7^x = 7^{-1} \longrightarrow \fbox{x=-1}"></p> <p><span class="spip-puce ltr"><b>–</b></span> <img src='https://www.matematicasies.com/local/cache-TeX/53eefa4b36d9f32d16bcd3f24c3fdf0f.png' style="vertical-align:middle;" width="285" height="65" alt="7^x = \frac{1}{49} \longrightarrow 7^x = 7^{-2} \longrightarrow \fbox{x=-2}" title="7^x = \frac{1}{49} \longrightarrow 7^x = 7^{-2} \longrightarrow \fbox{x=-2}"></p>