Solve for x log of x-2+ log of 9-x<1

Problem

(log_)(x−2)+(log_)(9−x)<1

Solution

1. Identify the domain of the logarithmic functions by ensuring the arguments are positive.

Domain: *x−2>0⇒x>2

Domain: *9−x>0⇒x<9

Combined Domain: *2<x<9

2. Apply the product rule for logarithms, (log_)(a)+(log_)(b)=(log_)(a*b) assuming base 10.

(log_)((x−2)*(9−x))<1

3. Rewrite the inequality in exponential form by raising the base (10) to both sides.

(x−2)*(9−x)<10

4. Expand the product on the left side of the inequality.

−x2+11*x−18<10

5. Rearrange into a quadratic inequality by moving all terms to one side.

−x2+11*x−28<0

6. Multiply by -1 and reverse the inequality sign to make the leading coefficient positive.

x2−11*x+28>0

7. Factor the quadratic expression.

(x−4)*(x−7)>0

8. Determine the solution intervals for the quadratic inequality.

x<4* or *x>7

9. Intersect the solution with the combined domain 2<x<9

(2<x<4)∪(7<x<9)

Final Answer

(log_)(x−2)+(log_)(9−x)<1⇒x∈(2,4)∪(7,9)

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