To solve \((0.528)^{\frac{2}{3}}\) using logarithms, you can apply the logarithm power rule and exponentiation. Here's how it works:

### Step 1: Take the natural logarithm (ln) of both sides.
Let \( y = (0.528)^{\frac{2}{3}} \). Then, take the natural logarithm on both sides:
\[
\ln(y) = \ln\left((0.528)^{\frac{2}{3}}\right)
\]

### Step 2: Apply the logarithm power rule.
The logarithm power rule states \(\ln(a^b) = b \cdot \ln(a)\). So:
\[
\ln(y) = \frac{2}{3} \cdot \ln(0.528)
\]

### Step 3: Compute \(\ln(0.528)\).
Using a calculator or logarithm table:
\[
\ln(0.528) \approx -0.638
\]

### Step 4: Multiply by \(\frac{2}{3}\).
\[
\ln(y) = \frac{2}{3} \cdot (-0.638) \approx -0.4253
\]

### Step 5: Exponentiate to solve for \(y\).
\[
y = e^{-0.4253} \approx 0.6535
\]

Thus, \((0.528)^{\frac{2}{3}} \approx 0.6535\) using logarithms.

Would you like to go over any specific steps in detail or have further questions?

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Here are 8 related questions:

1. How do you calculate \(\ln(0.4)\)?
2. What is \((0.8)^{\frac{2}{5}}\) using logarithms?
3. How do you solve \(\log(2.5^{\frac{3}{4}})\)?
4. What is \(\ln\left(2^{\frac{1}{3}}\right)\)?
5. How do you simplify \(\ln(0.5^{\frac{4}{3}})\)?
6. Can you find \(10^{\log(3.5)}\)?
7. What is the value of \((4.5)^{\frac{2}{3}}\) using logarithms?
8. How do you use logarithms to solve \(e^{\ln(0.25)}\)?

**Tip:** Logarithms are a powerful tool for simplifying calculations, especially when dealing with exponents.

Learn how to calculate (0.528)^(2/3) using logarithms with a step-by-step solution. This problem involves applying the logarithm power rule to simplify exponentiation. Suitable for advanced high school students or anyone interested in logarithmic calculations.

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