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

The answer to x^{ln(x)}=ex is x=e^{(1+sqrt(5))/2},x=e^{(1-sqrt(5))/2}

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x^{ln(x)}=ex

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

x = e^{\frac{1 + 5}{2}}, x = e^{\frac{1 - 5}{2}}

Decimal Notation

x = 5.04316 \dots, x = 0.53900 \dots

Solution steps

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

Apply log rules

ln (x) ln (x) = ln (e) + ln (x)

Rewrite the equation with

ln (x) = u

uu = ln (e) + u

Solve

uu = ln (e) + u : u = \frac{1 + 5}{2}, u = \frac{1 - 5}{2}

u = \frac{1 + 5}{2}, u = \frac{1 - 5}{2}

Substitute back

u = ln (x),

solve for

x

Solve

ln (x) = \frac{1 + 5}{2} : x = e^{\frac{1 + 5}{2}}

Solve

ln (x) = \frac{1 - 5}{2} : x = e^{\frac{1 - 5}{2}}

x = e^{\frac{1 + 5}{2}}, x = e^{\frac{1 - 5}{2}}

Verify Solutions:

x = e^{\frac{1 + 5}{2}}

True

, x = e^{\frac{1 - 5}{2}}

True

The solutions are

x = e^{\frac{1 + 5}{2}}, x = e^{\frac{1 - 5}{2}}

lo g_{5} (x + 3) = 2

151 = 10 lo g_{10} (x)

lo g_{x} (8) = 1.5

5 ln (2 x + 5) - 14 = 16

lo g_{3} (2 X - 6) = 1

What is the answer to x^{ln(x)}=ex ?
The answer to x^{ln(x)}=ex is x=e^{(1+sqrt(5))/2},x=e^{(1-sqrt(5))/2}