To solve the equation [tex]\( 5^{2y} = 625 \)[/tex], we will use the properties of exponents and logarithms.
1. Rewrite 625 as a power of 5:
We know that:
[tex]\[
625 = 5^4
\][/tex]
So, we can rewrite the equation as:
[tex]\[
5^{2y} = 5^4
\][/tex]
2. Use the property of exponents that states if [tex]\( a^m = a^n \)[/tex], then [tex]\( m = n \)[/tex]:
Since the bases are the same (5), we can set the exponents equal to each other:
[tex]\[
2y = 4
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
Divide both sides of the equation by 2:
[tex]\[
y = \frac{4}{2} = 2
\][/tex]
Therefore, the solution to the equation [tex]\( 5^{2y} = 625 \)[/tex] is:
[tex]\[
y = 2
\][/tex]
Thus, the correct answer is:
a. [tex]\(\{2\}\)[/tex]
