To determine how much of a log Kate can use to fill the remaining \( \frac{1}{3} \) meter, we first note that each log is \( \frac{2}{3} \) meter long.
We need to find what fraction of one log corresponds to \( \frac{1}{3} \) meter. We set up a proportion to do this:
\[ \text{Length of one log} = \frac{2}{3} \text{ meter} \]
Let \( x \) be the fraction of the log needed to fill \( \frac{1}{3} \) meter. Then we have the equation:
\[ x \cdot \frac{2}{3} = \frac{1}{3} \]
To solve for \( x \), we can multiply both sides of the equation by \( 3 \):
\[ 3 \cdot x \cdot \frac{2}{3} = 3 \cdot \frac{1}{3} \]
This simplifies to:
\[ 2x = 1 \]
Now, dividing both sides by \( 2 \):
\[ x = \frac{1}{2} \]
Thus, Kate can use \( \frac{1}{2} \) of a log to fill the remaining \( \frac{1}{3} \) meter.
The final answer is:
\[ \frac{1}{2} \]