Question

solutia naturala a ecuatiei $2^{x-5}+ 2^{x-3} + 3^{x-3} = 2^{x} + 3^{x-5}$ este:
A:3 B:11 C:8 D:5 E:15

Answer 1

2^x-5+2^x-3+3^x-3=2^x+3^x-5 este 8 deoarece:
[tex]2^x * 2^{-5} +2^x * 2^{-3} +3^x*3^{-3} =2^x+3^x*3^{-5} 2^x* \frac{1}{2^5} +2^x* \frac{1}{2^3} +3^x* \frac{1}{3^3} =2^x+3^x* \frac{1}{3^5} 2^x* \frac{1}{32} +2^x* \frac{1}{8} +3^x* \frac{1}{27} =2^x+3^x* \frac{1}{243} [/tex]
[tex] \frac{1}{32} + \frac{1}{8} +( \frac{3}{2} )^x* \frac{1}{27} =1+( \frac{3}{2} )^x* \frac{1}{243} [/tex]
$\frac{1}{27} *( \frac{3}{2} )^x- \frac{1}{243}*( \frac{3}{2} )^2=1- \frac{1}{32} - \frac{1}{8} = \frac{8}{243} *( \frac{3}{2})^x= \frac{27}{32}$
[tex]( \frac{3}{2} )^x = \frac{6561}{256} [/tex]
[tex]( \frac{3}{2} )^x = ( \frac{3}{2} )^8 deci x=8[/tex]