Sure, let's solve the equation [tex]\( 6 \cdot 2^{8t} = 24 \)[/tex] step-by-step.
1. Original Equation:
[tex]\[
6 \cdot 2^{8t} = 24
\][/tex]
2. Isolate the exponential term:
Divide both sides of the equation by 6:
[tex]\[
2^{8t} = \frac{24}{6}
\][/tex]
Simplifying the right-hand side:
[tex]\[
2^{8t} = 4
\][/tex]
3. Express 4 as a power of 2:
Note that [tex]\( 4 = 2^2 \)[/tex]:
[tex]\[
2^{8t} = 2^2
\][/tex]
4. Equate the exponents:
Since the bases are the same, the exponents must be equal:
[tex]\[
8t = 2
\][/tex]
5. Solve for [tex]\( t \)[/tex]:
Divide both sides of the equation by 8:
[tex]\[
t = \frac{2}{8}
\][/tex]
Simplify the fraction:
[tex]\[
t = \frac{1}{4}
\][/tex]
6. Express [tex]\( t \)[/tex] as a logarithm:
Recall that [tex]\( 2^{8t} = 2^2 \)[/tex] can be expressed in logarithmic form:
[tex]\[
8t = \log_2(4)
\][/tex]
Therefore,
[tex]\[
t = \frac{\log_2(4)}{8}
\][/tex]
So, the detailed step-by-step solution to the equation [tex]\( 6 \cdot 2^{8t} = 24 \)[/tex] expressed as a logarithm is:
[tex]\[
t = \frac{\log_2(4)}{8}
\][/tex]
Therefore, the correct answer is:
[tex]\( t = \frac{\log_2(4)}{8} \)[/tex].
