Solve for x log base 5 of x+ log base 5 of x-4=1

Problem

(log_5)(x)+(log_5)(x−4)=1

Solution

1. Apply the product rule for logarithms, which states that (log_b)(M)+(log_b)(N)=(log_b)(M*N)

(log_5)(x*(x−4))=1

2. Rewrite in exponential form by using the definition of a logarithm (log_b)(y)=a⇔ba=y

x*(x−4)=5

3. Distribute and rearrange the equation into a standard quadratic form a*x2+b*x+c=0

x2−4*x=5

x2−4*x−5=0

4. Factor the quadratic equation to find the potential values for x

(x−5)*(x+1)=0

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

x=5

x=−1

6. Check for extraneous solutions by ensuring the arguments of the original logarithms are positive. Since (log_5)(−1) and (log_5)(−1−4) are undefined, we discard x=−1

x=5

Final Answer

(log_5)(x)+(log_5)(x−4)=1⇒x=5

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