To solve the equation [tex]\(\log (2t + 4) = \log (14 - 3t)\)[/tex], we can follow these steps:
1. Understand the properties of logarithms: We know that if [tex]\(\log a = \log b\)[/tex], then [tex]\(a = b\)[/tex] provided that the bases of the logarithms are the same and the arguments [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are positive.
2. Remove the logarithms: Given [tex]\(\log (2t + 4) = \log (14 - 3t)\)[/tex], we can equate the arguments directly:
[tex]\[
2t + 4 = 14 - 3t
\][/tex]
3. Isolate the variable [tex]\(t\)[/tex]:
- Combine like terms involving [tex]\(t\)[/tex] on one side:
[tex]\[
2t + 3t + 4 = 14
\][/tex]
[tex]\[
5t + 4 = 14
\][/tex]
- Subtract 4 from both sides:
[tex]\[
5t = 10
\][/tex]
- Divide both sides by 5:
[tex]\[
t = 2
\][/tex]
4. Check for extraneous solutions: Logarithms are only defined for positive arguments. Thus,
[tex]\[
2t + 4 > 0 \quad \text{and} \quad 14 - 3t > 0.
\][/tex]
Checking these conditions with [tex]\(t = 2\)[/tex]:
- [tex]\(2(2) + 4 = 4 + 4 = 8\)[/tex] which is positive.
- [tex]\(14 - 3(2) = 14 - 6 = 8\)[/tex] which is also positive.
Therefore, [tex]\(t = 2\)[/tex] satisfies both conditions, being within the domain of the logarithmic functions.
Thus, the solution to the equation [tex]\(\log (2t + 4) = \log (14 - 3t)\)[/tex] is:
[tex]\[
t = 2
\][/tex]
