What is the answer to (ln(x))^2=2ln(x)+3 ?

The answer to (ln(x))^2=2ln(x)+3 is x=e^3,x= 1/e

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(ln(x))^2=2ln(x)+3

(ln (x))^{2} = 2 ln (x) + 3

x = e^{3}, x = \frac{1}{e}

Decimal

x = 20.08553 \dots, x = 0.36787 \dots

Solution steps

(ln (x))^{2} = 2 ln (x) + 3

Rewrite the equation with

ln (x) = u

(u)^{2} = 2 u + 3

Solve

u^{2} = 2 u + 3 : u = 3, u = - 1

u = 3, u = - 1

Substitute back

u = ln (x),

solve for

x

Solve

ln (x) = 3 : x = e^{3}

Solve

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

x = e^{3}, x = \frac{1}{e}

Verify Solutions:

x = e^{3}

True

, x = \frac{1}{e}

True

The solutions are

x = e^{3}, x = \frac{1}{e}

lo g_{6} (y) = 3

lo g_{10} (2 x) = 4

ln (10 x) + 1 = 0

lo g_{3} (x + 4) = - 4

lo g_{2} (x^{2} - 2 x - 27) = 3

What is the answer to (ln(x))^2=2ln(x)+3 ?
The answer to (ln(x))^2=2ln(x)+3 is x=e^3,x= 1/e
Solve for x: (ln(x))^2=2ln(x)+3
The solution is x=e^3,x= 1/e