Question

Resolva a equação: 5^x+5^x+1=3^x+3^x+1+3^x+2

Answer 1

$\mathrm{5^x+5^{x+1}=3^x+3^{x+1}+3^{x+2}}\\\\ \mathrm{5^x+5^x.5=3^x+3^x.3+3^x.9}\\\\ \mathrm{6.5^x=13.3^x\ \to\ \dfrac{5^x}{3^x}=\dfrac{13}{6}\ \to\ \bigg(\dfrac{5}{3}\bigg)^x=\dfrac{13}{6}}\\\\ \mathrm{\log{\bigg(\dfrac{5}{3}\bigg)^x}=\log{\bigg(\dfrac{13}{6}\bigg)}\ \to\ x\log{\bigg(\dfrac{5}{3}\bigg)}=\log{\bigg(\dfrac{13}{6}\bigg)}}\\\\ \mathrm{x=\dfrac{\log{\bigg(\dfrac{13}{6}\bigg)}}{\log{\bigg(\dfrac{5}{3}\bigg)}}\ \to\ \boxed{\mathbf{x=\log_{(\frac{5}{3})}{\bigg(\dfrac{13}{6}\bigg)}}}}$