Solve for x log of x+ log of x-9=1

Problem

(log_)(x)+(log_)(x−9)=1

Solution

1. Apply the product rule for logarithms, which states that (log_)(a)+(log_)(b)=(log_)(a*b)

(log_)(x*(x−9))=1

2. Rewrite in exponential form by using the base of the logarithm (base 10).

x*(x−9)=10

3. Expand the expression on the left side of the equation.

x2−9*x=10

4. Set the quadratic equation to zero by subtracting 10 from both sides.

x2−9*x−10=0

5. Factor the quadratic expression.

(x−10)*(x+1)=0

6. Solve for x by setting each factor equal to zero.

x=10

x=−1

7. Check for extraneous solutions by ensuring the arguments of the original logarithms are positive. Since (log_)(−1) and (log_)(−1−9) are undefined, x=−1 is excluded.

x=10

Final Answer

(log_)(x)+(log_)(x−9)=1⇒x=10

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