when two logs are divided they can be written thus:
x= log4 - log6
Take the log of both sides (it doesn't matter what base you use)
One of the properties of logs is that if they are the log of a number to a power, they are the same as the power times the log of the number. Meaning so
Divide both sides by log(6)
Take log base 6 of both sides
Use change of base formula to rewrite in a format you can plug into a calculator. Change of base formula says This works for any value of c. Typically the value chosen is 10 or e, as log base ten and log base e (natural log) are the two most common logs, and are programmed into any decent calculator.
Which is not exact, and I would seem to require a calculator to find.
It seems most likely to me that they are looking for or as these values are exact, and one of the other requirements was that he could not use a calculator.