Solve for x log of x-2+ log of x+5=2 log of 3

Problem

(log_)(x−2)+(log_)(x+5)=2*(log_)(3)

Solution

1. Apply the product rule for logarithms on the left side of the equation, which states (log_)(a)+(log_)(b)=(log_)(a*b)

(log_)((x−2)*(x+5))=2*(log_)(3)

2. Apply the power rule for logarithms on the right side of the equation, which states n*(log_)(a)=(log_)(an)

(log_)((x−2)*(x+5))=(log_)(3)

3. Remove the logarithms by using the property that if (log_)(u)=(log_)(v) then u=v

(x−2)*(x+5)=9

4. Expand the product on the left side using the FOIL method.

x2+5*x−2*x−10=9

5. Simplify and rearrange the equation into standard quadratic form a*x2+b*x+c=0 by subtracting 9 from both sides.

x2+3*x−19=0

6. Apply the quadratic formula x=(−b±√(,b2−4*a*c))/(2*a) where a=1 b=3 and c=−19

x=(−3±√(,3−4*(1)*(−19)))/(2*(1))

7. Simplify the discriminant and the expression.

x=(−3±√(,9+76))/2

x=(−3±√(,85))/2

8. Check for extraneous solutions by ensuring the arguments of the original logarithms are positive. Since √(,85)≈9.22 the value x=(−3−√(,85))/2 is negative, which would make x−2 negative. Thus, only the positive root is valid.

x=(−3+√(,85))/2

Final Answer

(log_)(x−2)+(log_)(x+5)=2*(log_)(3)⇒x=(−3+√(,85))/2

Want more problems? Check here!