Solve for x log of x+ log of x-3=1

Problem

(log_)(x)+(log_)(x−3)=1

Solution

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

(log_)(x*(x−3))=1

2. Rewrite in exponential form by using the definition of a logarithm with base 10.

x*(x−3)=10

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

x2−3*x=10

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

x2−3*x−10=0

5. Factor the quadratic expression by finding two numbers that multiply to −10 and add to −3

(x−5)*(x+2)=0

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

x=5

x=−2

7. Check for extraneous solutions by ensuring the arguments of the original logarithms are positive. Since (log_)(−2) and (log_)(−2−3) are undefined, we discard x=−2

x=5

Final Answer

(log_)(x)+(log_)(x−3)=1⇒x=5

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