If you like that table, we can adapt it to the change of base rule (although I'm going to put it in the form $$ \log_2 x = log_2 4 \times \log_4 x $$ and use this table

$$\begin{array}{rrrrrrrrrr} \text{exponent}& 0& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10\\ \text{power of 2}&1& 2& 4& 8& 16& 32& 64& 128& 256& 512& 1024\\ \text{power of 4}&1& 4& 16& 64& 256& 1024& 4096& 16384& 65536& 262144& 1048576 \end{array}$$

The top row, labeled exponents, could also be labeled 'logarithms' as you noted, and they can be to the base $2$ or the base $4$ depending on which lower row you are looking at.

Imagine that there are counters in the bottom two rows, and they both start in the first column, on the $1$. Start moving the bottom counter forward, one column at a time. Now suppose the 'powers of 2' counter starts moving forward and it tries to stay on the same number as the 'powers of 4' counter is on. So when 'powers of 4' jumps to '4', 'powers of 2' has to move two columns to get to its 4. When 'powers of 4' then moves to 16, 'powers of 2' again has to move two columns to get to its 16. I.e. the powers of 2 counter has to move twice as fast as the powers of 4 counter. That means when you read off the logarithms from the top row $$\log_2 x = 2 \times \log_4x.$$

If you added a 'powers of 8' row you could easily see that $\log_2 = 3 \times \log_8x$, and then generalize it to compare $\log_a$ and $\log_{a^n}$.

Motivated by @dxiv's comment, we can adapt your table to understand the change of base rule, although I'm just going to illustrate changing base from $2$ to $4$ and prove the formula in the (equivalent) form $$ \log_2 x = log_2 4 \times \log_4 x. $$ Let's add a row to your table:

$$\begin{array}{rrrrrrrrrr} \text{exponent}& 0& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10\\ \text{power of 2}&1& 2& 4& 8& 16& 32& 64& 128& 256& 512& 1024\\ \text{power of 4}&1& 4& 16& 64& 256& 1024& 4096& 16384& 65536& 262144& 1048576 \end{array}$$

The top row, labeled exponents, is where you read off the logarithms, and they can be to the base $2$ or the base $4$ depending on which lower row you are looking at.

Imagine that there are counters in the bottom two rows, and they both start in the first column, on their $1$. Start moving the bottom counter forward, one column at a time. Now suppose the 'powers of 2' counter starts moving forward and it tries to stay on the same number as the 'powers of 4' counter is on. So when 'powers of 4' jumps to '4', 'powers of 2' has to move two columns to get to its 4. When 'powers of 4' then moves to 16, 'powers of 2' again has to move two columns to get to its 16. I.e. the powers of 2 counter has to move twice as fast as the powers of 4 counter. That means when you read off the logarithms from the top row $$\log_2 x = 2 \times \log_4x.$$

If you added a 'powers of 8' row you could easily see that $\log_2 = 3 \times \log_8x$, and then generalize it to compare $\log_a$ and $\log_{a^n}$.