Question

Resolver el siguiente logaritmo

log3*(x^2)+log3(x-6)=0

Answer 1

Expandimos el logaritmo
$\log _{10}\left(3\right)x^2+\log _{10}\left(3\right)\left(x-6\right)$
$=x\log _{10}\left(3\right)-6\log _{10}\left(3\right)$
$=\log _{10}\left(3\right)x^2+x\log _{10}\left(3\right)-6\log _{10}\left(3\right)$
Nos da una ecuación cuadrática
$x^2\log _{10}\left(3\right)+x\log _{10}\left(3\right)-6\log _{10}\left(3\right)=0$
Se puede resolver de varias maneras, lo haré con la fórmula cuadrática
Esta dicta que:
$\quad x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$
Nos da que este logaritomo es
$x=\frac{-\frac{\ln \left(3\right)}{\ln \left(10\right)}+\sqrt{\left(\frac{\ln \left(3\right)}{\ln \left(10\right)}\right)^2-4\left(-\frac{6\ln \left(3\right)}{\ln \left(10\right)}\right)\frac{\ln \left(3\right)}{\ln \left(10\right)}}}{2\frac{\ln \left(3\right)}{\ln \left(10\right)}}=2$
o
$x=\frac{-\frac{\ln \left(3\right)}{\ln \left(10\right)}-\sqrt{\left(\frac{\ln \left(3\right)}{\ln \left(10\right)}\right)^2-4\left(-\frac{6\ln \left(3\right)}{\ln \left(10\right)}\right)\frac{\ln \left(3\right)}{\ln \left(10\right)}}}{2\frac{\ln \left(3\right)}{\ln \left(10\right)}}=-3$
x=-3, x=2