The answer to your first problem is that a^ and "log to the base a" are inverse functions. So a^{log(a)x} = x and log(a)a^x = x by definition.

ln(5) + ln(10) = ln(x)

The sum of two logs multiplies the arguments, ie ln(a) + ln(b) = ln(ab), so

ln(5*10) = ln(x)

ln(50) = ln(x)

Thus x = 50.

-Dan