Question

Resolva cada uma das equações exponenciais a seguir.
Utilize as aproximações log2= 0,301 e log3= 0,477.
a)2^{x} =12
b)5^{x} =60
c)1,5^{x-1} =6
d)2^{ \frac{-t}{5700} =100

Answer 1

Explicação passo-a-passo:

a)

$\sf 2^x=12$

$\sf x=log_{2}~12$

$\sf x=\dfrac{log~12}{log~2}$

$\sf x=\dfrac{log~(2^2\cdot3)}{log~2}$

$\sf x=\dfrac{log~2^2+log~3}{log~2}$

$\sf x=\dfrac{2\cdot log~2+log~3}{log~2}$

$\sf x=\dfrac{2\cdot0,301+0,477}{0,301}$

$\sf x=\dfrac{0,602+0,477}{0,301}$

$\sf x=\dfrac{1,079}{0,301}$

$\sf x=\dfrac{1079}{301}$

$\sf \red{x=3,585}$

b)

$\sf 5^x=60$

$\sf x=log_{5}~60$

$\sf x=\dfrac{log~60}{log~5}$

$\sf x=\dfrac{log~(10\cdot2\cdot3)}{log~\Big(\frac{10}{2}\Big)}$

$\sf x=\dfrac{log~10+log~2+log~3}{log~10-log~2}$

$\sf x=\dfrac{1+0,301+0,477}{1-0,301}$

$\sf x=\dfrac{1,778}{0,699}$

$\sf x=\dfrac{1778}{699}$

$\sf \red{x=2,544}$

c)

$\sf 1,5^{x-1}=6$

$\sf x-1=log_{1,5}~6$

$\sf x-1=\dfrac{log~6}{log~1,5}$

$\sf x-1=\dfrac{log~(2\cdot3)}{log~\Big(\frac{3}{2}\Big)}$

$\sf x-1=\dfrac{log~2+log~3}{log~3-log~2}$

$\sf x-1=\dfrac{0,301+0,477}{0,477-0,301}$

$\sf x-1=\dfrac{0,778}{0,176}$

$\sf x-1=\dfrac{778}{176}$

$\sf x-1=4,42$

$\sf x=4,42+1$

$\sf \red{x=5,42}$

d)

$\sf 2^{\frac{-t}{5700}}=100$

$\sf \dfrac{-t}{5700}=log_{2}~100$

$\sf \dfrac{-t}{5700}=\dfrac{log~100}{log~2}$

$\sf \dfrac{-t}{5700}=\dfrac{2}{0,301}$

$\sf \dfrac{-t}{5700}=\dfrac{2000}{301}$

$\sf \dfrac{-t}{5700}=6,6445$

$\sf -t=5700\cdot6,6445$

$\sf -t=37873,65$

$\sf \red{t=-37873,65}$