To simplify [tex]\(\log_5(625)\)[/tex] without a calculator, let's break it down step-by-step.
1. Understand the Expression: The problem is asking for the logarithm of 625 with base 5. In mathematical terms, [tex]\(\log_5(625)\)[/tex] asks the question: "To what power should we raise 5 to obtain 625?"
2. Express 625 as a Power of 5: We need to see if 625 can be written as [tex]\(5^n\)[/tex] for some integer [tex]\(n\)[/tex].
Let's try to find such [tex]\(n\)[/tex]:
- [tex]\(5^1 = 5\)[/tex]
- [tex]\(5^2 = 25\)[/tex]
- [tex]\(5^3 = 125\)[/tex]
- [tex]\(5^4 = 625\)[/tex]
We see that [tex]\(625 = 5^4\)[/tex].
3. Apply the Property of Logarithms: Given [tex]\(625 = 5^4\)[/tex], we can use the property of logarithms which states that [tex]\(\log_b(b^k) = k\)[/tex] when [tex]\(b\)[/tex] is the base and [tex]\(k\)[/tex] is the exponent.
So, [tex]\(\log_5(625) = \log_5(5^4)\)[/tex].
4. Simplify: Using the logarithmic property mentioned, [tex]\(\log_5(5^4) = 4\)[/tex].
Therefore, the simplified form of [tex]\(\log_5(625)\)[/tex] is:
[tex]\[
\boxed{4}
\][/tex]
