Answer :

To solve the equation [tex]\(625^x = 3125\)[/tex], let's break it down step by step.

1. Express each number as a power of 5:
- [tex]\(625\)[/tex] can be expressed as a power of 5:
[tex]\[
625 = 5^4
\][/tex]
- [tex]\(3125\)[/tex] can also be expressed as a power of 5:
[tex]\[
3125 = 5^5
\][/tex]

2. Rewrite the original equation using these powers of 5:
- Substitute these expressions into the equation:
[tex]\[
(5^4)^x = 5^5
\][/tex]

3. Simplify the left side using the power of a power property:
- According to the power of a power property, [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. So:
[tex]\[
5^{4x} = 5^5
\][/tex]

4. Equate the exponents because the bases are the same:
- If [tex]\(5^{4x} = 5^5\)[/tex], then the exponents must be equal:
[tex]\[
4x = 5
\][/tex]

5. Solve for [tex]\(x\)[/tex]:
- Divide both sides by 4 to isolate [tex]\(x\)[/tex]:
[tex]\[
x = \frac{5}{4}
\][/tex]
- Which simplifies to:
[tex]\[
x = 1.25
\][/tex]

Therefore, the solution for [tex]\(x\)[/tex] is [tex]\(1.25\)[/tex].

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