log_a(a ^ 2) - 4 * log_a(1/a) where a > 0 and a 1

We are given the expression:

\[
\log_a(a^2) - 4 \cdot \log_a\left(\frac{1}{a}\right)
\]
where \( a > 0 \) and \( a \neq 1 \).

Let's simplify it step by step.

### Step 1: Simplify \( \log_a(a^2) \)
Using the logarithmic property \( \log_b(b^x) = x \), we can simplify \( \log_a(a^2) \):
\[
\log_a(a^2) = 2
\]

### Step 2: Simplify \( \log_a\left(\frac{1}{a}\right) \)
We know the property \( \log_b\left(\frac{1}{b}\right) = -1 \), so:
\[
\log_a\left(\frac{1}{a}\right) = -1
\]

### Step 3: Apply the result to the entire expression
Substitute the simplified values back into the original expression:
\[
2 - 4 \cdot (-1)
\]

### Step 4: Simplify further
\[
2 - 4 \cdot (-1) = 2 + 4 = 6
\]

Thus, the value of the expression is:

\[
\boxed{6}
\]

Would you like more details or have any questions?

Here are some related questions to explore:
1. What is the general formula for logarithmic properties?
2. How do logarithmic identities simplify more complex expressions?
3. What happens to logarithmic functions when the base is changed?
4. How does logarithmic differentiation work for variable exponents?
5. How would the expression change if \( a = 1 \)?

**Tip:** Always remember the key logarithmic identities for simplifying expressions, like \( \log_b(b^x) = x \) and \( \log_b(1/b) = -1 \).

log_a(a^2) - 4 * log_a(1/a) where a > 0 and a ≠ 1

Learn how to simplify the logarithmic expression log_a(a^2) - 4 * log_a(1/a), where a > 0 and a ≠ 1. This problem explores key logarithmic identities, including the use of log_b(b^x) = x and log_b(1/b) = -1, making it ideal for students in Grades 10-12.

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