Solve for x: x^(lg^3 x-5*lgx)-0.0001=0

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll shift 0.0001 to the right side:

x^(lg^3 x-5*lgx) = 0.0001

We'll take decimal logarithms both sides:

lg [x^((lg x)^3-5*lgx)] = lg 0.0001

We'll use the power property of logarithms:

((lg x)^3 -5*lgx)*lg x = lg 10^(-4)

We'll remove the brackets:

(lg x)^4 - 5*(lg x)^2 = -4

(lg x)^4 - 5*(lg x)^2 + 4 = 0

We'll replace (lg x)^2 by t:

t^2 - 5t + 4 = 0

t1 = 4 and t2 = 1

(lg x)^2 = t1 <=> (lg x)^2=4 => lg x = 2 or lg x = -2

lg x = 2 => x = 10^2 => x = 100

lg x = -2 => x = 10^(-2) = 1/100 = 0.01

(lg x)^2 = t2 <=> (lg x)^2=1 => lg x = 1 or lg x = -1

lg x = 1 => x = 10

lg x = -1 => x = 10^(-1) => x = 1/10 => x = 0.1

The solutions of the equation are: {0.01 , 0.1 , 10 , 100}.

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