Question

1) Resolva as equações exponenciais:
a) 2x = 8
b) 2x = 64
c) 25x = 125
d) 7x = 343
e) 2x = 1/32

2) Resolva as equações exponenciais:
a) 4x = 64
b) (1/2)x= (1/32)
c) 2x+1 = 1
d) (2/3)x = 8/27
e) 8x+1 = 128x-3

Answer 1

Explicação passo-a-passo:

1)

a)

$\sf 2^x=8$

$\sf 2^x=2^3$

Igualando os expoentes:

$\sf \red{x=3}$

b)

$\sf 2^x=64$

$\sf 2^x=2^6$

Igualando os expoentes:

$\sf \red{x=6}$

c)

$\sf 25^x=125$

$\sf (5^2)^x=5^3$

$\sf 5^{2x}=5^3$

Igualando os expoentes:

$\sf 2x=3$

$\sf \red{x=\dfrac{3}{2}}$

d)

$\sf 7^x=343$

$\sf 7^x=7^3$

Igualando os expoentes:

$\sf \red{x=3}$

e)

$\sf 2^x=\dfrac{1}{32}$

$\sf 2^x=\dfrac{1}{2^5}$

$\sf 2^x=2^{-5}$

Igualando os expoentes:

$\sf \red{x=-5}$

2)

a)

$\sf 4^x=64$

$\sf 4^x=4^3$

Igualando os expoentes:

$\sf \red{x=3}$

b)

$\sf \Big(\dfrac{1}{2}\Big)^x=\dfrac{1}{32}$

$\sf \Big(\dfrac{1}{2}\Big)^x=\Big(\dfrac{1}{2}\Big)^5$

Igualando os expoentes:

$\sf \red{x=5}$

c)

$\sf 2^{x+1}=1$

$\sf 2^{x+1}=2^0$

Igualando os expoentes:

$\sf x+1=0$

$\sf \red{x=-1}$

d)

$\sf \Big(\dfrac{2}{3}\Big)^x=\dfrac{8}{27}$

$\sf \Big(\dfrac{2}{3}\Big)^x=\Big(\dfrac{2}{3}\Big)^3$

Igualando os expoentes:

$\sf \red{x=3}$

e)

$\sf 8^{x+1}=128^{x-3}$

$\sf (2^3)^{x+1}=(2^7)^{x-3}$

$\sf 2^{3x+3}=2^{7x-21}$

Igualando os expoentes:

$\sf 3x+3=7x-21$

$\sf 3x-7x=-21-3$

$\sf -4x=-24~~~~\cdot(-1)$

$\sf 4x=24$

$\sf x=\dfrac{24}{4}$

$\sf \red{x=6}$