To solve the equation [tex]\(\log (t - 3) = \log (17 - 4t)\)[/tex], follow these detailed steps:
1. Understand the property of logarithms:
The equation [tex]\(\log (a) = \log (b)\)[/tex] implies that [tex]\(a = b\)[/tex], provided [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are both positive.
2. Apply the property:
Given [tex]\(\log (t - 3) = \log (17 - 4t)\)[/tex], we can set the arguments of the logarithms equal to each other:
[tex]\[
t - 3 = 17 - 4t
\][/tex]
3. Solve for [tex]\(t\)[/tex]:
- Start by isolating [tex]\(t\)[/tex] on one side. Add [tex]\(4t\)[/tex] to both sides to combine the [tex]\(t\)[/tex] terms:
[tex]\[
t + 4t - 3 = 17
\][/tex]
This simplifies to:
[tex]\[
5t - 3 = 17
\][/tex]
- Next, add 3 to both sides to isolate the term with [tex]\(t\)[/tex]:
[tex]\[
5t - 3 + 3 = 17 + 3
\][/tex]
This simplifies to:
[tex]\[
5t = 20
\][/tex]
- Finally, divide by 5 to solve for [tex]\(t\)[/tex]:
[tex]\[
t = \frac{20}{5} = 4
\][/tex]
4. Check the solution:
For the solution [tex]\(t = 4\)[/tex] to be valid, the arguments of the original logarithms must be positive:
- Check [tex]\(t - 3 > 0\)[/tex]:
[tex]\[
4 - 3 = 1 > 0
\][/tex]
- Check [tex]\(17 - 4t > 0\)[/tex]:
[tex]\[
17 - 4 \times 4 = 17 - 16 = 1 > 0
\][/tex]
Both conditions are satisfied, so the solution [tex]\(t = 4\)[/tex] is valid.
Therefore, the solution to the equation [tex]\(\log (t - 3) = \log (17 - 4t)\)[/tex] is [tex]\(\boxed{4}\)[/tex].
