Question #3cf9a

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Answer 1 · 2017-07-25T06:53:19

#x = frac(ln(32 pm 12 sqrt(7)))(2 ln(2))#

We have: #4^(2 x) + (1 - 65) (4^(x)) + 16 = 0#

#Rightarrow (4^(x))^(2) - 64 (4^(x)) + 16 = 0#

#Rightarrow 4^(x) = frac(64 pm sqrt((- 64)^(2) - 4(1)(16)))(2(1))#

#Rightarrow 4^(x) = frac(64 pm sqrt(4096 - 64))(2)#

#Rightarrow 4^(x) = frac(64 pm sqrt(4032))(2)#

#Rightarrow 4^(x) = frac(64 pm 24 sqrt(7))(2)#

#Rightarrow 4^(x) = 32 pm 12 sqrt(7)#

Then, let's apply #ln# to both sides of the equation:

#Rightarrow ln(4^(x)) = ln(32 pm 12 sqrt(7))#

Using the laws of logarithms:

#Rightarrow x ln(4) = ln(32 pm 12 sqrt(7))#

#Rightarrow x ln(2^(2)) = ln(32 pm 12 sqrt(7))#

#Rightarrow x = frac(ln(32 pm 12 sqrt(7)))(ln(2^(2)))#

#therefore x = frac(ln(32 pm 12 sqrt(7)))(2 ln(2))#