Solve for x natural log of x^2-12 = natural log of x

Problem

ln(x2−12)=ln(x)

Solution

1. Apply the one-to-one property of logarithms, which states that if ln(a)=ln(b) then a=b

x2−12=x

2. Rearrange the equation into standard quadratic form a*x2+b*x+c=0 by subtracting x from both sides.

x2−x−12=0

3. Factor the quadratic expression by finding two numbers that multiply to −12 and add to −1

(x−4)*(x+3)=0

4. Solve for x by setting each factor equal to zero using the zero product property.

x−4=0⇒x=4

x+3=0⇒x=−3

5. Check for extraneous solutions by ensuring the arguments of the original logarithms are positive.

For *x=4:4−12=4>0* and *4>0

For *x=−3:(−3)2−12=−3<0* and −3<0

Since the natural log of a negative number is undefined, x=−3 is extraneous.

Final Answer

ln(x2−12)=ln(x)⇒x=4

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