Question

Equação Exponencial D) E) F)

Answer 1

d)

$5^{ \frac{3}{2}x} = 125^4 \\ \\ 5^{ \frac{3}{2}x} = (5^3)^4 \\ \\ 5^{ \frac{3}{2}x} = 5^{12} \\ \\ \frac{3}{2}x = 12 \\ \\ x = \frac{12\cdot2}{3} \\ \\ \boxed{x = 8}$

E)

$\sqrt[6]{1000} \,\,\cdot\,10^x = 0,\!001 \\ \\ 1000^{ \frac{1}{6}} \,\,\cdot\,\,10^x = 10^{-3} \\ \\ (10^3)^{ \frac{1}{6}} \,\,\cdot\,\,10^x=10^{-3} \\ \\ 10^{ \frac{3}{6}} \,\,\cdot\,\,10^x = 10^{-3} \\ \\ 10^{ \frac{3}{6}+x} = 10^{-3} \\ \\ \frac{3}{6} +x = -3 \\ \\ \frac{1}{2}+x = -3 \\ \\ x = -3- \frac{1}{2} \\ \\ \boxed{x= - \frac{7}{2} }$

F)

$\frac{8^x}{2^{x+2}} = 256 \\ \\ \frac{(2^3)^x}{2^{x+2}} = 2^8 \\ \\ \frac{2^{3x}}{2^{x+2}} = 2^8 \\ \\ 2^{3x - (x +2)} = 2^8\\ \\ 2^{2x -2} = 2^8 \\ \\ 2x-2 = 8 \\ \\ 2x = 10 \\ \\ \boxed{x = 5}$