Solve for $t$: $\log (t-3)=\log (17-4t)$.

• A. No real solution (the logarithm arguments cannot both be positive).
• B. $t=2$
• C. $t=4$
• D. $t=5$

Answer: (C)

Verified step by step guidance

1. Recognize that the equation involves logarithms with the same base and is given as \(\log(t - 3) = \log(17 - 4t)\). Since the logarithm function is one-to-one, if \(\log A = \log B\), then \(A = B\), provided both \(A\) and \(B\) are positive.
2. Set the arguments of the logarithms equal to each other: \(t - 3 = 17 - 4t\).
3. Solve the resulting linear equation for \(t\): add \$4t\( to both sides and add \)3\( to both sides to isolate \)t$ terms on one side and constants on the other.
4. After finding the value(s) of \(t\), check the domain restrictions for the logarithms. Both arguments, \(t - 3\) and \$17 - 4t$, must be greater than zero to be valid inputs for the logarithm function.
5. Verify which solution(s) satisfy the domain restrictions. Discard any solution that makes either argument non-positive, and identify the valid solution(s) as the final answer.