Simply solve for x ?
4ˣ - 2ˣ - 1 = 0
4ˣ - 2ˣ - 1 = 0
........
Favorite Answer
4ˣ - 2ˣ - 1 = 0
(2ˣ)² - 2ˣ - 1 = 0
2ˣ = {1 ± √(1 + 4) } / 2
2ˣ = (1±√5)/2
(1-√5)/2 is negative, so use (1+√5)/2
ln(2ˣ) = ln{(1+√5)/2}
x*ln(2) = ln{(1+√5)/2}
x = ln{(1+√5)/2} / ln(2)
= { ln(1+√5) - ln(2) } / ln(2)
= ln(1+√5) / ln(2) - 1
x=.694241913630617301738790266898
I checked the answer, and it works..
Anonymous
4ˣ - 2ˣ - 1 = 0 ...... the initial equation
(2²)ˣ - 2ˣ - 1 = 0 ...... attempting to change to a consistent base
2²ˣ - 2ˣ - 1 = 0 ..... simplified
(2ˣ)² - 2ˣ - 1 = 0 .... changed to same base, and into quadratic form
(2ˣ)² - 2ˣ = 1 .... moved the constant to one side
(2ˣ)² - 2ˣ + ¼ = 1 + ¼ .... complete the square
(2ˣ - ½)² = 5/4 .... turn into binomial squared
2ˣ - ½ = ±√(5/4) ..... square root both sides
2ˣ - ½ = ±√5 / 2 ..... denominator in radical
2ˣ = ½ ± √5 / 2 ..... isolate x
2ˣ = (1 ± √5) / 2 .... common denominator
log 2ˣ = log [(1 ± √5) / 2] ... isolate x by logarithms
x log 2 = log [(1 ± √5) / 2] .... exponent in logarithm rule
x = log [(1 ± √5) / 2] / log 2 .... division to isolate x
This is as accurate and all encompassing a solution as you can get.
Working out specific values...
x = 0.6942419... (real)
x = -0.6942419... + i 4.5323601... (complex)
Tubby
4ˣ - 2ˣ - 1 = 0
2^2x - 2^x - 2^0 = 0
:> 2x - x - 0 = 0
x = 0
rui_min
maybe x is equal to 1/6.
a_ebnlhaitham
4ˣ - 2ˣ - 1 = 2ˣ - 1 = 0
2ˣ = 1
ˣ = 0.5