To solve the given equation step-by-step, arrange the steps in the following order:
1. Add 4 to each side of the equation:
[tex]\[
3\left(2^{2 t-5}\right) = 14
\][/tex]
2. Divide both sides of the equation by 3:
[tex]\[
2^{2 t-5} = \frac{14}{3}
\][/tex]
3. Take the log of each side:
[tex]\[
\log \left(2^{2 t-5}\right) = \log \left(\frac{14}{3}\right)
\][/tex]
4. Use the Exponential Property and write [tex]\(\frac{14}{3}\)[/tex] in decimal form:
[tex]\[
(2 t-5) \log 2 = \log 4.67
\][/tex]
5. Divide each side by [tex]\(\log 2\)[/tex]:
[tex]\[
2 t-5 = \frac{\log 4.67}{\log 2}
\][/tex]
6. Find the value of [tex]\(\frac{\log 4.67}{\log 2}\)[/tex] and substitute:
[tex]\[
2 t-5 = 2.23
\][/tex]
7. Add 5 to each side of the equation:
[tex]\[
2 t = 2.23 + 5
\][/tex]
8. Simplify:
[tex]\[
t \approx 3.625
\][/tex]
So, the final solution is [tex]\( t \approx 3.625 \)[/tex].
