Question

Resolva as equações exponenciais:

A) 3^x-2 =9
B) 10^1-x =1/10
C) 5^x²-2x =125
D) 2^-x²+4x =8
E) 3^x-2 =1/27

Answer 1

$a) 3^{ x-2 } =9\\\\3^{ x-2 } = 3^{2}\\ \\x-2=2\\\\x=2+2\\\\x=4$

$b) 10^{1-x} = \frac{1}{10} \\\\ 10^{1-x} = 10^{-1}\\\\1-x=-1\\\\-x=-1-1\\\\-x=-2\\\\ x=2$

$c) 5^{ x^{2} -2x} =125 \\\\ 5^{ x^{2} -2x} = 5^{3} \\ \\ x^{2} -2x=3 \\\\ x^{2} -2x-3=0 \\\\\\ \Delta= b^{2} -4.a.c\\ \Delta=(- 2)^{2} -4.1.(-3) \\ \Delta=4+12 \\ \Delta=16\\\\ \\ x= \frac{-b+- \sqrt{\Delta} }{2.a} \\\\ x= \frac{-(-2)+- \sqrt{16} }{2.1} \\\\ x= \frac{4+-4}{2}$
$x'= \frac{4+4}{2}\\x'= \frac{8}{2} \\x'=4 \\ \\ x''= \frac{4-4}{2}\\x''= \frac{0}{2} \\x''=0 \\ \\ \\ \\ S=(0,4)$

$d) 2^{ x^{2} +4x} =8 \\\\ 2^{ x^{2} +4x} = 2^{3} \\\\ x^{2} +4x=3\\\\x^{2} +4x-3=0\\\\ \Delta= 4^{2} -4.1.(-3)\\ \Delta=16+12 \\ \Delta=28 \\\\\\ x= \frac{-4+- \sqrt{28} }{2.1} \\\\ x= \frac{-4+-5,3}{2} \\\\ x'=\frac{-4+5,3}{2}\\ \\x'= \frac{1,3}{2}\\\\\\x''= \frac{-4-5,3}{2}\\\\ x''= \frac{-9,3}{2}\\\\\\\\S=( -\frac{9}{3} , \frac{1,3}{2} )$

$e)3^{x-2} = \frac{1}{27} \\\\ 3^{x-2} = 27^{-1} \\\\ 3^{x-2} = 3^{3.(-1)} \\x-2=3.(-1)\\\\x-2=-3\\\\x=-3+2\\\\x=1$