ln(x-5)-ln(5x-21)=ln(1/x)

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Solution for ln(x-5)-ln(5x-21)=ln(1/x) equation:


D( x )

x = 0

1/x <= 0

5*x-21 <= 0

x-5 <= 0

x = 0

x = 0

1/x <= 0

1/x <= 0

1*x^-1 <= 0

1*x^-1 <= 0 // : 1

x^-1 <= 0/1

x^-1 <= 0

1/(x^1) <= 0

x <> 0

1/(x^1) <= 0 // * x^2

(x^2)/(x^1) <= 0

x^1 <= 0

x <= 0

x in (-oo:0)

5*x-21 <= 0

5*x-21 <= 0

5*x-21 <= 0 // + 21

5*x <= 21 // : 5

x <= 21/5

x-5 <= 0

x-5 <= 0

x-5 <= 0 // + 5

x <= 5

x in (5:+oo)

ln(x-5)-ln(5*x-21) = ln(1/x) // - ln(1/x)

ln(x-5)-ln(5*x-21)-ln(1/x) = 0

ln(1*(x-5))-ln(5*x-21)-ln(1/x) = 0

ln((1*(x-5))/(5*x-21))-ln(1/x) = 0

ln(((1*(x-5))/(5*x-21))/(1/x)) = 0

ln(((x-5)/(5*x-21))/(1/x)) = 0

ln(((x-5)/(5*x-21))/(1/x)) = ln(e^0)

((x-5)/(5*x-21))/(1/x) = e^0

((x-5)/(5*x-21))/(1/x)-e^0 = 0

(x-5)/(x^-1*(5*x-21))-1 = 0

(x-5)/(x^-1*(5*x-21))+(-1*x^-1*(5*x-21))/(x^-1*(5*x-21)) = 0

x-1*x^-1*(5*x-21)-5 = 0

x+21*x^-1-10 = 0

x+21*x^-1-10 = 0

1*x^1+21*x^-1-10*x^0 = 0

(1*x^2-10*x^1+21*x^0)/(x^1) = 0 // * x^2

x^1*(1*x^2-10*x^1+21*x^0) = 0

x^1

x^2-10*x+21 = 0

x^2-10*x+21 = 0

DELTA = (-10)^2-(1*4*21)

DELTA = 16

DELTA > 0

x = (16^(1/2)+10)/(1*2) or x = (10-16^(1/2))/(1*2)

x = 7 or x = 3

x in { 3, 7}

(x-3)*(x-7) = 0

((x-3)*(x-7))/(x^-1*(5*x-21)) = 0

((x-3)*(x-7))/(x^-1*(5*x-21)) = 0 // * x^-1*(5*x-21)

(x-3)*(x-7) = 0

( x-3 )

x-3 = 0 // + 3

x = 3

( x-7 )

x-7 = 0 // + 7

x = 7

x in { 3}

x = 7

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