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 using the base of the common logarithm, which is 10

x*(x+15)=10

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

x2+15*x=100

4. Set the equation to zero by subtracting 100 from both sides.

x2+15*x−100=0

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

(x+20)*(x−5)=0

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

x=−20

x=5

7. Check for extraneous solutions by ensuring the arguments of the original logarithms are positive. Since (log_)(−20) is undefined, we discard x=−20

x=5

Final Answer

(log_)(x)+(log_)(x+15)=2⇒x=5

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