Solve for x log base 5 of x-5+ log base 5 of x+15=3

Problem

(log_5)(x−5)+(log_5)(x+15)=3

Solution

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

(log_5)((x−5)*(x+15))=3

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

(x−5)*(x+15)=5

3. Expand the product on the left side and evaluate the power on the right side.

x2+15*x−5*x−75=125

4. Combine like terms and set the quadratic equation to zero by subtracting 125 from both sides.

x2+10*x−200=0

5. Factor the quadratic expression by finding two numbers that multiply to −200 and add to 10

(x+20)*(x−10)=0

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

x=−20

x=10

7. Check for extraneous solutions by ensuring the arguments of the original logarithms are positive. For x=−20 the term x−5 becomes −25 which is undefined. For x=10 both 10 - 5 = 5a*n*d0 + 15 = 25$ are positive.

x=10

Final Answer

(log_5)(x−5)+(log_5)(x+15)=3⇒x=10

Want more problems? Check here!