Sure, let's solve the logarithmic equation [tex]\(\log_{0.2} 625 = H\)[/tex].
### Step-by-Step Solution:
1. Understand the meaning of the logarithm:
The equation [tex]\(\log_{0.2} 625 = H\)[/tex] translates to the question: "To what power must the base 0.2 be raised to obtain 625?"
2. Change of base formula:
To solve a logarithm with a base other than 10 or [tex]\(e\)[/tex], it is often convenient to use the change of base formula:
[tex]\[
\log_b a = \frac{\log_k a}{\log_k b}
\][/tex]
where [tex]\(k\)[/tex] can be any positive number (commonly 10 or [tex]\(e\)[/tex]). In this case, we can use natural logarithms (base [tex]\(e\)[/tex]) or common logarithms (base 10), but let's just generalize it here for simplicity.
3. Apply the change of base:
Using the change of base formula, we get:
[tex]\[
\log_{0.2} 625 = \frac{\log 625}{\log 0.2}
\][/tex]
4. Evaluate the logarithms:
Evaluate the values on the right-hand side using a calculator:
[tex]\[
\log 625 \quad \text{and} \quad \log 0.2
\][/tex]
5. Calculate the value:
After evaluating these, you find that:
[tex]\[
\frac{\log 625}{\log 0.2} = -4
\][/tex]
So, the value of [tex]\(H\)[/tex] is:
[tex]\[
H = -4
\][/tex]
Therefore, the solution to [tex]\(\log_{0.2} 625\)[/tex] is [tex]\(-4\)[/tex].
