we note that most handheld calculators have button marked logbut this is typically for calculating the logarithm base- not base-two computing the logarithm function exactly for any integer involves the use of calculusbut we can use an approximation that is good enough for our purposes without calculus in particularwe can easily compute the smallest integer greater than or equal to logb (its so-called ceilinglogb for positive integernthis value is equal to the number of times we can divide by before we get number less than or equal to for examplethe evaluation of log is because (( / )/ )/ likewiselog is because (( / )/ )/ and log is because ((( / )/ )/ )/ < the following proposition describes several important identities that involve logarithms for any base greater than proposition (logarithm rules)given real numbers and we have logb (aclogb logb logb ( /clogb logb logb (ac logb logb logd alogd blogd alogd by conventionthe unparenthesized notation lognc denotes the value log(nc we use notational shorthandlogc nto denote the quantity(log ) in which the result of the logarithm is raised to power the above identities can be derived from converse rules for exponentiation that we will present on page we illustrate these identities with few examples example we demonstrate below some interesting applications of the logarithm rules from proposition (using the usual convention that the base of logarithm is if it is omittedlog( nlog log log nby rule log( / log log log by rule log log nby rule log log nby rule log (log )log (log )/ by rule log nlog nby rule as practical matterwe note that rule gives us way to compute the base-two logarithm on calculator that has base- logarithm buttonlogfor log log log