Insert Means in an Arithmetic Sequence (that contains logarithms)

Question

So the question is:

You have an Arithmetic Sequence. Log 2 and Log 1024 are two terms in the sequence

Find 8 arithmetic means between them.

Answer 1

Here's a hint. Suppose we wanted to find a single arithmetic mean between $\log a$ and $\log b$. This would be $$\frac12\left(\log a + \log b\right)$$ which, using properties of logarithms, is equal to $\log\sqrt{ab}$. For example, the arithmetic mean of $\log 2$ and $\log 8$ is $\log\sqrt{2\cdot 8} = \log 4$, and $\log 2, \log 4,$ and $\log 8$ form an arithmetic sequence of three terms. (Perhaps you might like to check this with your calculator.)

Now could you do the same thing with more than one mean? If it's not clear how to do eight, could you possibly do two?

(George Pólya says “If you can't solve this problem, is there a simpler problem of the same type that you can solve?”)