Question

solve. simplify your answer(s). if there are multiple answers, separate them with commas. \\(\log_{3}(-5u + 1) = \log_{3}(-u + 13)\\) \\(u = \square\\)

Explanation:

Step1: Use the property of logarithms

If \(\log_{a}b=\log_{a}c\), then \(b = c\) (for \(a>0,a
eq1,b>0,c>0\)). So we set \(-5u + 1=-u + 13\).

Step2: Solve the linear equation

Subtract \(-u\) from both sides: \(-5u+u + 1=-u+u + 13\), which simplifies to \(-4u + 1=13\).
Subtract 1 from both sides: \(-4u+1 - 1=13 - 1\), so \(-4u = 12\).
Divide both sides by \(-4\): \(\frac{-4u}{-4}=\frac{12}{-4}\), giving \(u=-3\).
We should check the domain of the logarithmic functions. For \(\log_{3}(-5u + 1)\), when \(u = - 3\), \(-5(-3)+1=15 + 1=16>0\). For \(\log_{3}(-u + 13)\), when \(u=-3\), \(-(-3)+13=3 + 13=16>0\). So the solution is valid.

Answer:

\(-3\)