The expression in the image is $3(1 + \log_3 2)$. Let's solve it step-by-step.

Step 1: Simplify the expression

We know the formula for logarithmic calculations: $\log_a a^x = x$ Here, the logarithm is to base 3 ($\log_3$), and the argument is 2. So, let's rewrite:

$3(1 + \log_3 2)$

Step 2: Calculate each term inside the parentheses

1. $1$ remains as it is.
2. $\log_3 2$: Use the change of base formula to calculate it: $\log_3 2 = \frac{\log 2}{\log 3}$ Use approximate values: $\log 2 \approx 0.3010 \quad \text{and} \quad \log 3 \approx 0.4771$ So: $\log_3 2 = \frac{0.3010}{0.4771} \approx 0.631$

Step 3: Substitute back into the expression

Now, substitute $\log_3 2 \approx 0.631$ into the expression: $3(1 + 0.631)$ Simplify: $3(1.631) = 4.893$

Final Answer:

The value of $3(1 + \log_3 2)$ is approximately 4.893.

Would you like further clarifications or more details? 😊

Here are 5 related questions to deepen your understanding:

1. What is the change of base formula in logarithms, and how is it derived?
2. Can the value of $\log_3 2$ be solved exactly without using approximations?
3. How can you verify the accuracy of $\log_3 2$ using a calculator?
4. What happens if we change the base of the logarithm to 10 or $e$?
5. How would this problem change if the expression were $3(1 - \log_3 2)$?

Tip: When solving logarithms, knowing key logarithmic properties, like the change of base formula, can make calculations much simpler!