Solve for x log of x+ log of x+21=2

Problem

(log_)(x)+(log_)(x+21)=2

Solution

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

(log_)(x*(x+21))=2

2. Rewrite in exponential form by using the base of the common logarithm, which is 10

x*(x+21)=10

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

x2+21*x=100

x2+21*x−100=0

4. Factor the quadratic expression by finding two numbers that multiply to −100 and add to 21

(x+25)*(x−4)=0

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

x=−25

x=4

6. Check for extraneous solutions by ensuring the arguments of the original logarithms are positive. Since (log_)(−25) is undefined, we discard x=−25

x=4

Final Answer

(log_)(x)+(log_)(x+21)=2⇒x=4

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