Historical note: Logarithms were first introduced in order to multiply and raise powers and roots efficiently, and not just as "the inverse function of the exponent" (which is significant for some reason...)
So our motivating question is: How can one multiply efficiently.
But then we need to go back, and show a geometric progression (and this should also be motivated). For example:
1, 2, 4, 8, 16, 32, 64, 128
Then ask: Can we multiply numbers that are in this table efficiently? For example, what is 4 * 8? It is 32 which is already in the table.
Try to find a method for doing this efficiently and discover that the index of the product is the sum of the indices.
And then ask: What if we want to multiply 3 by 9? Well we could create a progression with quotient 3 instead.
But how do we multiply 2 by 3?
Then we can make a progression with quotient 1.0001 or something like that, and then build this huge table (just like the real BOOK of logarithms that used to be sold) and this would allow us to multiply.
Also, either now or before, show how this table can be used to powers and roots.
Then you can also ask, can you create some mechanical device that does this without a huge book? And then discover the slide rule.