Question

final review
score: 12/100 answered: 3/17
question 5
solve the equation \\(\log_{3}(x + 25) + \log_{3}(x + 1) = 4\\).
if there are two solutions, separate them by a comma. if there is no solution, enter \dne\.
\\(x = \square\\)

Explanation:

Step1: Apply log addition rule

Using the property \(\log_a M+\log_a N = \log_a(MN)\), we get \(\log_3[(x + 25)(x + 1)]=4\).

Step2: Convert to exponential form

Since \(\log_a b=c\) implies \(a^c = b\), we have \(3^4=(x + 25)(x + 1)\).

Step3: Simplify and solve quadratic

Calculate \(3^4 = 81\), so \(x^2+26x + 25=81\).
Rearrange to \(x^2+26x - 56 = 0\).
Factor (or use quadratic formula): \(x^2+28x - 2x - 56 = 0\), \(x(x + 28)-2(x + 28)=0\), \((x - 2)(x + 28)=0\).
Solutions: \(x = 2\) or \(x=-28\).

Step4: Check domain

For \(\log_3(x + 25)\) and \(\log_3(x + 1)\), \(x+25>0\) and \(x + 1>0\), so \(x>-1\).
\(x=-28\) is invalid, \(x = 2\) is valid.

Answer:

\(2\)