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## Sagot :

First, recall the properties of exponents. We have:

[tex]\[ f(x) = 2 \cdot (5)^{2x} \][/tex]

We can rewrite the exponent in a different form by using exponentiation properties. Specifically, [tex]\(a^{b \cdot c} = (a^b)^c\)[/tex]. Here, we can set [tex]\(a = 5\)[/tex] and [tex]\(b = 2x\)[/tex]:

[tex]\[ (5)^{2x} = [(5^2)]^x \][/tex]

Next, we calculate [tex]\(5^2\)[/tex]:

[tex]\[ 5^2 = 25 \][/tex]

Now substitute [tex]\(25\)[/tex] back into the function:

[tex]\[ f(x) = 2 \cdot (25)^x \][/tex]

So, the function [tex]\( f(x) = 2(5)^{2x} \)[/tex] is equivalent to [tex]\( f(x) = 2(25)^x \)[/tex].

From the given options:

1. [tex]\(f(x) = 50^x\)[/tex]

2. [tex]\(f(x) = 100^x\)[/tex]

3. [tex]\(f(x) = 2(25)^x\)[/tex]

4. [tex]\(f(x) = 4(25)^x\)[/tex]

The equivalent function to [tex]\( f(x) = 2(5)^{2x} \)[/tex] is:

[tex]\[ f(x) = 2(25)^x \][/tex]

Thus, the correct option is:

[tex]\[ \boxed{3} \][/tex]