Solving 10^x = 100 is quite easy. But how do mathematicians solve something like 10^x = 128?

They discovered that the answer x to this problem 10^x = 1+N can be got by calculating an infinite sum:

If N is between –1 and 1 to find x, add up the following infinite list of numbers:
c.(N-N²/2+N³/3-…)

This infinite series of numbers will converge to some finite number provided N is between –1 and 1. The number c here changes according to which base you are taking. For base 10, c is approximately 1/2.3. Using this formula allowed mathematicians to prepare tables of logarithms.
If N is bigger than 1 then we use the following trick:
Log(1+N)= c.(N-N²/2+N³/3-…)
-Log(1-N)= c.(N-N²/2+N³/3-…)

Hence
Log((1+N)/(1-N))=Log(1+N)-Log(1-N)=2c(N+ N³/3+…) (*)
But now to calculate Log(1+M) for M bigger than 1 we just have to find an N between –1 and 1 with (1+N)/(1-N)=1+M and then use the series (*).

Why is the base that Gauss's prime number dice favours so nice? Because in this case c=1 in this formula. It is the reason that this logarithm to base e is called the "natural logarithm".

Why were logarithms such an advantage to merchants and sailors in the seventeenth and eighteenth century?

If you had to multiply 1673 by 1364, that would take time.

Instead merchants would do the following:
(1) Using their tables, find the logarithms of 1673 and 1364.
(2) Add these two logarithms together (much simpler than doing multiplication)
(3) Now reverse the logarithm tables to find the number whose logarithm is the number you get in step 2.

Why does this work? Because 10^x x 10^y= 10^x+y the connection between multiplication and addition again - the heart of the logarithm function.

Laplace, some 200 years after Napier's death, declared that his table of logarithms "by shortening the labours, doubled the life of the astronomer".