To solve the equation [tex]\(1.5 \cdot 4^{2x} = 12\)[/tex], follow these steps:
1. Isolate the exponential expression:
[tex]\[
4^{2x} = \frac{12}{1.5}
\][/tex]
Simplify the fraction:
[tex]\[
4^{2x} = 8
\][/tex]
2. Express both sides with base 2 (since 4 and 8 are powers of 2):
[tex]\[
(2^2)^{2x} = 2^3
\][/tex]
3. Simplify the left-hand side:
[tex]\[
2^{4x} = 2^3
\][/tex]
4. Since the bases are the same, set the exponents equal to each other:
[tex]\[
4x = 3
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{3}{4}
\][/tex]
The value of [tex]\(x\)[/tex] is [tex]\(\frac{3}{4}\)[/tex] or 0.75 when written as a decimal.
6. Round to the nearest hundredth (if needed):
The value of [tex]\(x\)[/tex] is [tex]\(0.75\)[/tex].
Thus, the value of [tex]\(x\)[/tex] that satisfies the equation is [tex]\(\boxed{0.75}\)[/tex].
