What is the answer to ln^2(x)-ln(x)-2=0 ?

The answer to ln^2(x)-ln(x)-2=0 is x=e^2,x= 1/e

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ln^2(x)-ln(x)-2=0

ln^{2} (x) - ln (x) - 2 = 0

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

Decimal

x = 7.38905 \dots, x = 0.36787 \dots

Solution steps

ln^{2} (x) - ln (x) - 2 = 0

Rewrite the equation with

ln (x) = u

u^{2} - u - 2 = 0

Solve

u^{2} - u - 2 = 0 : u = 2, u = - 1

u = 2, u = - 1

Substitute back

u = ln (x),

solve for

x

Solve

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

Solve

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

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

Verify Solutions:

x = e^{2}

True

, x = \frac{1}{e}

True

The solutions are

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

20 lo g_{10} (x) = 60

lo g_{5} (4) + lo g_{5} (x) = 0

2 lo g_{10} (x) - 2 lo g_{10} (x + 1) = 0

lo g_{3} (2 x - 1) = 1

3 = - lo g_{10} (x)

What is the answer to ln^2(x)-ln(x)-2=0 ?
The answer to ln^2(x)-ln(x)-2=0 is x=e^2,x= 1/e
Solve for x: ln^2(x)-ln(x)-2=0
The solution is x=e^2,x= 1/e