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so previously weve discovered that the integral of the natural log of x is equal to x by the natural log of x minus x plus an integration constant and ill link a video in the description or the top right hand corner if you want to know how we got to that result but what if we were to more generally extend this to the integral of log to the base a of x so log to the any base of x as long as a is a positive real number unfortunately i dont know how to do this but i am going to manipulate my way to a result that i do know which weve already established the pathway that ill get there is to use the change of base rule for logarithms so you might recall from your earlier studies of logarithms that log to the base a of x can be expressed as log and now we can choose any base lets choose the base e so log to the base a of x is equal to log to the base e of x divided by log to the base e of a okay so this is the change of base rule and of course now log to the base e is simply the natura