Find all real solutions of the equation ln(x^4)+ln(x^2)-ln(x^3)-2=7?

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The equation to be solved is: ln(x^4) + ln(x^2) - ln(x^3) - 2 = 7

ln(x^4) + ln(x^2) - ln(x^3) - 2 = 7

use the property of logarithms that gives: log a^b = b*log a

=> 4*ln x + 2*ln x - 3*ln x = 7 + 2

=> 3*ln x = 9

=> ln x = 9/3

=> ln x = 3

The base of ln is e. This gives x = e^3

The required solution of the equation is x = e^3

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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First, we'll use the power property of logarithms:

4ln x + 2ln x - 3ln x - 2 = 7

We'll keep to the left side all the terms that contain logarithms:

3ln x = 7 + 2

3 ln x = 9

We'll divide by 3 both sides:

ln x = 3

x = e^3

The solution of the given equation is x = e^3.

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