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Let's solve each of the logarithmic expressions one by one:

### a. \( \log_5 0.04 \)
We can write \( 0.04 \) as a fraction:
\[
0.04 = \frac{1}{25} = \frac{1}{5^2}
\]
Thus, the expression becomes:
\[
\log_5 \left( \frac{1}{5^2} \right) = \log_5 (5^{-2})
\]
Using the logarithmic rule \( \log_b (b^x) = x \), we get:
\[
\log_5 (5^{-2}) = -2
\]
So, the result is \( -2 \).

### b. \( \log_{\sqrt{3}} 27 \)
First, express \( \sqrt{3} \) as \( 3^{1/2} \) and \( 27 \) as \( 3^3 \), so the expression becomes:
\[
\log_{3^{1/2}} (3^3)
\]
Using the logarithmic rule \( \log_{b^c} (x) = \frac{1}{c} \log_b (x) \), this becomes:
\[
\frac{1}{1/2} \log_3 (3^3) = 2 \cdot 3 = 6
\]
Thus, the result is \( 6 \).

### c. \( \log_4 \sqrt{2} \)
First, express \( \sqrt{2} \) as \( 2^{1/2} \), so the expression becomes:
\[
\log_4 (2^{1/2})
\]
Next, express \( 4 \) as \( 2^2 \), and apply the logarithmic rule \( \log_{b^c} (x) = \frac{1}{c} \log_b (x) \):
\[
\frac{1}{2} \log_2 (2^{1/2}) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}
\]
Thus, the result is \( \frac{1}{4} \).

### d. \( \log_6 \left( \frac{1}{216} \right) \)
First, express \( 216 \) as \( 6^3 \), so the expression becomes:
\[
\log_6 \left( \frac{1}{6^3} \right) = \log_6 (6^{-3})
\]
Using the logarithmic rule \( \log_b (b^x) = x \), we get:
\[
\log_6 (6^{-3}) = -3
\]
Thus, the result is \( -3 \).

### e. \( \log_{81} \left( \frac{1}{3} \right) \)
First, express \( 81 \) as \( 3^4 \), so the expression becomes:
\[
\log_{3^4} \left( \frac{1}{3} \right) = \log_{3^4} (3^{-1})
\]
Using the logarithmic rule \( \log_{b^c} (x) = \frac{1}{c} \log_b (x) \):
\[
\frac{1}{4} \log_3 (3^{-1}) = \frac{1}{4} \cdot (-1) = -\frac{1}{4}
\]
Thus, the result is \( -\frac{1}{4} \).

### Summary:
a. \( -2 \)  
b. \( 6 \)  
c. \( \frac{1}{4} \)  
d. \( -3 \)  
e. \( -\frac{1}{4} \)

Would you like further details on any step or explanation? Here are 5 related questions:

1. How do logarithmic properties simplify complex expressions?
2. What happens when the base of a logarithm is a fraction?
3. How can we express roots in logarithmic equations?
4. Why is \( \log_b(b^x) = x \) a fundamental logarithmic identity?
5. How does changing the logarithmic base affect calculations?

**Tip:** Always try to express numbers as powers of the base in logarithmic equations—it simplifies the computation considerably.

Carilah nilai dari tiap logaritma berikut!

This page provides a detailed solution to a logarithmic problem set. Learn how to evaluate logarithms with different bases, including fractional and negative exponents. Suitable for high school students (Grades 10-12).

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