Simply solve for x ?

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..

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)

4ˣ - 2ˣ - 1 = 0

2^2x - 2^x - 2^0 = 0

:> 2x - x - 0 = 0

x = 0

maybe x is equal to 1/6.

4ˣ - 2ˣ - 1 = 2ˣ - 1 = 0

2ˣ = 1

ˣ = 0.5