What is the answer to (2x)^{ln(2x)}=e^2(2x) ?

The answer to (2x)^{ln(2x)}=e^2(2x) is x=(e^2)/2 ,x= 1/(2e)

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(2x)^{ln(2x)}=e^2(2x)

(2 x)^{l n (2 x)} = e^{2} (2 x)

x = \frac{e ^{2}}{2}, x = \frac{1}{2 e}

Decimal Notation

x = 3.69452 \dots, x = 0.18393 \dots

Solution steps

(2 x)^{l n (2 x)} = e^{2} (2 x)

Apply log rules

ln (2 x) ln (2 x) = ln (2 x) + 2

Rewrite the equation with

ln (2 x) = u

uu = u + 2

Solve

uu = u + 2 : u = 2, u = - 1

u = 2, u = - 1

Substitute back

u = ln (2 x),

solve for

x

Solve

ln (2 x) = 2 : x = \frac{e ^{2}}{2}

Solve

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

x = \frac{e ^{2}}{2}, x = \frac{1}{2 e}

Verify Solutions:

x = \frac{e ^{2}}{2}

True

, x = \frac{1}{2 e}

True

The solutions are

x = \frac{e ^{2}}{2}, x = \frac{1}{2 e}

lo g_{a} (y) + lo g_{a} (5) = 7

lo g_{10} (4 x + 8) = 2 + lo g_{10} (3)

ln (3) + ln (6 - x) = 2 ln (x)

0 = \frac{1 - 3 ln ( x )}{x ^{3}}

a = 20 lo g_{10} (x)

What is the answer to (2x)^{ln(2x)}=e^2(2x) ?
The answer to (2x)^{ln(2x)}=e^2(2x) is x=(e^2)/2 ,x= 1/(2e)