Help

[pliant drum]

I thought of enumerating, but I think there should be some cleverer ways. Help if you know how, thanks.

[crisp plover]

How many possible combinations are there?

Then, how many lead to duplicate values of x?

I get 55 possible x values.

[pliant drum]

But how do you find all duplicate values without enumerations

Or is it just impossible to not enumerate

[crisp plover]

the only duplicates that I see are:

$\bullet \quad \log_b{1} = 0\ \text{ for }\ b\in{2, 3, 4, ..., 9}$

$\bullet \quad \log_2{4} = \log_3{9} = 2$

$\bullet \quad \log_4{2} = \log_9{3} = \frac{1}{2}$

Can you see any others?

[deft knot]

Jay