Solve for x log of x+ log of x-15=2

Problem

(log_)(x)+(log_)(x−15)=2

Solution

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

(log_)(x*(x−15))=2

2. Rewrite in exponential form by assuming the base is 10 (common logarithm).

x*(x−15)=10

3. Expand and simplify the equation into a standard quadratic form.

x2−15*x=100

x2−15*x−100=0

4. Factor the quadratic equation by finding two numbers that multiply to −100 and add to −15

(x−20)*(x+5)=0

5. Solve for x by setting each factor to zero.

x=20

x=−5

6. Check for extraneous solutions by ensuring the arguments of the original logarithms are positive. Since (log_)(−5) is undefined, we discard x=−5

x=20

Final Answer

(log_)(x)+(log_)(x−15)=2⇒x=20

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