To solve the equation [tex]\(4(2.5)^{2x} = 4\)[/tex], we will follow these steps:
1. Divide both sides of the equation by 4:
[tex]\[
\frac{4(2.5)^{2x}}{4} = \frac{4}{4}
\][/tex]
Simplifying this, we get:
[tex]\[
(2.5)^{2x} = 1
\][/tex]
2. Take the logarithm of both sides. This will help us solve for [tex]\(x\)[/tex]:
[tex]\[
\log((2.5)^{2x}) = \log(1)
\][/tex]
3. Use the properties of logarithms to simplify. Recall that [tex]\(\log(a^b) = b \log(a)\)[/tex]:
[tex]\[
2x \log(2.5) = \log(1)
\][/tex]
4. Remember that [tex]\(\log(1) = 0\)[/tex]:
[tex]\[
2x \log(2.5) = 0
\][/tex]
5. Solve for [tex]\(x\)[/tex]. Divide both sides of the equation by [tex]\( \log(2.5) \)[/tex]:
[tex]\[
x = \frac{0}{2 \log(2.5)}
\][/tex]
Since any number divided by a non-zero number is 0, we get:
[tex]\[
x = 0
\][/tex]
Thus, the value of [tex]\(x\)[/tex] that satisfies the equation [tex]\(4(2.5)^{2x} = 4\)[/tex] is:
[tex]\[
\boxed{0}
\][/tex]
