What is the answer to 2^2n=4^n ?

The answer to 2^2n=4^n is n=1,n= 1/2

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2^2n=4^n

2^{2} n = 4^{n}

n = 1, n = \frac{1}{2}

Solution steps

2^{2} n = 4^{n}

Prepare

2^{2} n = 4^{n}

for Lambert form:

e^{l n (2) (2 - 2 n)} n = 1

Rewrite the equation with

- 2 n ln (2) = u

and

n = - \frac{u}{2 ln ( 2 )}

e^{l n (2) (2 - 2 (- \frac{u}{2 l n ( 2 )}))} (- \frac{u}{2 ln ( 2 )}) = 1

Simplify

- \frac{e ^{2 l n (2) + u} u}{2 ln ( 2 )} = 1

Rewrite

- \frac{e ^{2 l n (2) + u} u}{2 ln ( 2 )} = 1

in Lambert form:

e^{u} u = - \frac{ln ( 2 )}{2}

Solve

e^{u} u = - \frac{ln ( 2 )}{2} : u = - 2 ln (2), u = - ln (2)

u = - 2 ln (2), u = - ln (2)

Substitute back

u = - 2 n ln (2),

solve for

n

Solve

- 2 n ln (2) = - 2 ln (2) : n = 1

Solve

- 2 n ln (2) = - ln (2) : n = \frac{1}{2}

n = 1, n = \frac{1}{2}

e^{k} = 2

e^{x} = e^{- 4 x}

x, e^{e^{x}} = 10

4096 = 2^{n}

4096 = 2^{x}