To solve the equation [tex]\(3\left(2^{2t-5}\right) - 4 = 10\)[/tex], let's follow these steps in order:
1. Simplify the equation by isolating the exponential term:
[tex]\[
3\left(2^{2t - 5}\right) - 4 + 4 = 10 + 4
\][/tex]
Simplified:
[tex]\[
3\left(2^{2t - 5}\right) = 14
\][/tex]
2. Divide both sides by 3 to further isolate the exponential term:
[tex]\[
\left(2^{2t - 5}\right) = \frac{14}{3}
\][/tex]
3. Take the logarithm of each side to decompose the exponent:
[tex]\[
\log \left(2^{2t - 5}\right) = \log \left(\frac{14}{3}\right)
\][/tex]
4. Use the power rule of logarithms (i.e., [tex]\( \log(a^b) = b \log(a) \)[/tex]) to bring down the exponent:
[tex]\[
(2t - 5) \log(2) = \log \left(\frac{14}{3}\right)
\][/tex]
5. Solve for [tex]\( t \)[/tex] by isolating it step by step:
[tex]\[
2t - 5 = \frac{\log \left(\frac{14}{3}\right)}{\log(2)}
\][/tex]
Simplify further:
[tex]\[
2t = \frac{\log \left(\frac{14}{3}\right)}{\log(2)} + 5
\][/tex]
Finally:
[tex]\[
t = \frac{\frac{\log \left(\frac{14}{3}\right)}{\log(2)} + 5}{2}
\][/tex]
Given that we have the numerical solution involved in detailed logarithm calculations, the intermediate result for [tex]\( \frac{\log \left(\frac{14}{3}\right)}{\log(2)}\)[/tex] is approximately 2.722, and the final value of [tex]\( t \)[/tex] is approximately 3.611. Therefore,
Thus, following all steps correctly, we can approximate:
[tex]\[
t \approx 3.611
\][/tex]
