Solve for n

Problem

ln(n2+12)=ln(−9*n−2)

Solution

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

n2+12=−9*n−2

2. Rearrange the equation into standard quadratic form a*n2+b*n+c=0 by adding 9*n and 2 to both sides.

n2+9*n+14=0

3. Factor the quadratic expression by finding two numbers that multiply to 14 and add to 9

(n+7)*(n+2)=0

4. Solve for n by setting each factor equal to zero.

n+7=0⇒n=−7

n+2=0⇒n=−2

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

For *n=−7:(−7)2+12=61>0* and −9*(−7)−2=61>0

For *n=−2:(−2)2+12=16>0* and −9*(−2)−2=16>0

Final Answer

ln(n2+12)=ln(−9*n−2)⇒n=−7,−2

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