Solve for x

Problem

(log_5)(x)+(log_5)(x+1)=(log_5)(20)

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+1))=(log_5)(20)

2. Equate the arguments of the logarithms since the bases are the same and the function is one-to-one.

x*(x+1)=20

3. Expand the expression on the left side to form a quadratic equation.

x2+x=20

4. Rearrange the equation into standard quadratic form a*x2+b*x+c=0 by subtracting 20 from both sides.

x2+x−20=0

5. Factor the quadratic by finding two numbers that multiply to −20 and add to 1

(x+5)*(x−4)=0

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

x=−5

x=4

7. Check for extraneous solutions by ensuring the arguments of the original logarithms are positive. Since (log_5)(−5) is undefined, we discard x=−5

x=4

Final Answer

(log_5)(x)+(log_5)(x+1)=(log_5)(20)⇒x=4

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