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, which is 10 for a common logarithm.

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 by finding two numbers that multiply to −10 and add to 9

(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_)(−10) is undefined, we discard x=−10

x=1

Final Answer

(log_)(x)+(log_)(x+9)=1⇒x=1

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