To evaluate the expression:

$\frac{2}{3} \log_2(3)$

we can proceed as follows:

1. Interpretation of the Expression:

   • This expression is in the form of a constant $\frac{2}{3}$ multiplied by $\log_2(3)$, the logarithm of 3 with base 2.

2. Approximate Value:

   • First, we find $\log_2(3)$ approximately. Using the change of base formula:

     $\log_2(3) = \frac{\log_{10}(3)}{\log_{10}(2)}$

   • With approximate values for common logarithms: $\log_{10}(3) \approx 0.4771 \quad \text{and} \quad \log_{10}(2) \approx 0.3010$

     $\log_2(3) \approx \frac{0.4771}{0.3010} \approx 1.585$

3. Multiplying by $\frac{2}{3}$:

   • Now, we multiply by $\frac{2}{3}$:

     $\frac{2}{3} \times 1.585 \approx 1.057$

So, the approximate value of $\frac{2}{3} \log_2(3)$ is:

$\approx 1.057$

Would you like more details on any step, or do you have further questions?

---

1. How do we calculate logarithms with different bases without a calculator?
2. What is the significance of the base in logarithmic expressions?
3. How do we interpret the result of $\log_2(3)$ in terms of exponents?
4. How can we simplify expressions involving fractional constants in logarithms?
5. Can we express $\log_2(3)$ as a fraction?

Tip: Using the change of base formula can simplify many logarithmic calculations and help convert between bases efficiently.