solve the logarithmic equation. be sure to reject any value of x that is not in the domain of the original…

solve the logarithmic equation. be sure to reject any value of x that is not in the domain of the original logarithmic expressions. give the exact answer. \\(\\log_{5}(x - 3) + \\log_{5}(x + 121) = 3\\) select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. \\(x = \\square\\) (simplify your answer. use a comma to separate answers as needed.) b. there is no solution.

solve the logarithmic equation. be sure to reject any value of x that is not in the domain of the original logarithmic expressions. give the exact answer. \\(\\log_{5}(x - 3) + \\log_{5}(x + 121) = 3\\) select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. \\(x = \\square\\) (simplify your answer. use a comma to separate answers as needed.) b. there is no solution.

Answer

Explanation:

Step1: Apply log addition rule

Using the property (\log_b M + \log_b N=\log_b(MN)), we get (\log_5[(x - 3)(x + 121)] = 3).

Step2: Convert to exponential form

By the definition of logarithms, if (\log_b y = k), then (y = b^k). So, ((x - 3)(x + 121)=5^3 = 125).

Step3: Expand and simplify

Expand the left - hand side: (x^2+121x-3x - 363 = 125), which simplifies to (x^2 + 118x-363 - 125=0), or (x^2+118x - 488 = 0). We can use the quadratic formula (x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}) for the quadratic equation (ax^2+bx + c = 0). Here, (a = 1), (b = 118), (c=-488). First, calculate the discriminant (\Delta=b^2 - 4ac=(118)^2-4\times1\times(-488)=13924 + 1952=15876). (\sqrt{\Delta}=\sqrt{15876}=126). Then (x=\frac{-118\pm126}{2}). We have two solutions: (x_1=\frac{-118 + 126}{2}=\frac{8}{2}=4) and (x_2=\frac{-118 - 126}{2}=\frac{-244}{2}=-122).

Step4: Check the domain

For the original logarithmic equation (\log_5(x - 3)+\log_5(x + 121)=3), the arguments of the logarithms must be positive.

  • For (x=-122): (x - 3=-122-3=-125\lt0) and (x + 121=-122 + 121=-1\lt0), so (x=-122) is not in the domain.
  • For (x = 4): (x - 3=4 - 3 = 1\gt0) and (x + 121=4+121 = 125\gt0), so (x = 4) is in the domain.

Answer:

A. (x = 4)