(1.10)^x=2.4

1.10^x  =  2.4

If the variable is an exponent, find the log of both sides:

log(1.10^x)  =  log(2.4)

An exponent within a log comes out as a multiplier:

x·log(1.10)  =  log(2.4)

Divide both sides by log(1.10):

x  =  log(2.4) / log(1.10)

x  =  9.185 ...