Solve for [tex]\( x \)[/tex]:
[tex]\[ 3x = 6x - 2 \][/tex]



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[tex]\[\log_{0.2} 625 = H\][/tex]
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Response:
[tex]\[ \log_{0.2} 625 = H \][/tex]



Answer :

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].