Question

alguem me ajuda! eu imploro !!

determine o conjunto solução das equações exponenciais do tipo;

a) 4^x = 16
b) 25^x=√5
c) (1/9)^x=√27
d)49^2^x^+^3=343
e)(5/3)^x =27/125
f)(2^x)^x=8^x
g)17^x^2-^4^x=1

Answer 1

$a)\\4^x=16\\\\4^x=4^2\\\\x=2\\\\\\b)\\25^x=\sqrt{5}\\\\\left(5^2\right)^x=5^{\frac{1}{2}}\\\\5^{\;2\;.\;x}=5^{\frac{1}{2}}\\\\5^{\;2x}=5^{\frac{1}{2}}\\\\2x = \frac{1}{2}\\\\x=\frac{1}{4}$

$c)\\\left(\frac{1}{9}\right)^x=\sqrt{27}\\\\\left(9^{-1}\right)^x=27^{\frac{1}{2}}\\\\\left(\left(3^2\right)^{-1}\right)^x=\left(3^3\right)^{\frac{1}{2}}\\\\3^{\;2\;.\;-1\;.\;x}=3^{\;3\;.\;\frac{1}{2}}\\\\3^{-2x}=3^{\frac{3}{2}}\\\\-2x=\frac{3}{2}\\\\x=-\frac{3}{4}$

d) Creio que tu esqueceu de algo, da uma conferida.

$e)\\\left(\frac{5}{3}\right)^x=\frac{27}{125}\\\\\left(\frac{5}{3}\right)^x=\left(\frac{3^3}{5^3}\right)\\\\\left(\frac{5}{3}\right)^x=\left(\frac{3}{5}\right)^3\\\\\left(\frac{5}{3}\right)^x=\left( \left(\frac{5}{3}\right)^{-1}\right)^3\\\\\left(\frac{5}{3}\right)^x= \left(\frac{5}{3}\right)^{\;-1\;.\;3}\\\\\left(\frac{5}{3}\right)^x= \left(\frac{5}{3}\right)^{-3}\\\\x=-3$

$f)\\\left(2^x\right)^x=8^x\\\\2^{\;x\;.\;x}=\left(2^3\right)^x\\\\2^{x^2}=2^{\;3\;.\;x}\\\\x^2=3x\\\\x^2-3x=0\\\\x.(x-3)=0\;\;\;\rightarrow\;equacao\;segundo\;grau\;incompleta\\\\x'=0\\x''=3$

g) Está confusa, tente ajeitar utilizando chaves para separar e organizar os expoentes.