Derivative of logs

Ok so everyone knows that the derivative of ln x is 1/|x| . (if you don't you should... oh and applied calc students apparently learn it 1/x but have an x>=0 tagged at the end to make it legal... go figure!)

But I was talking to a calculus student the other day that was not sure about the derivative of log(base a) x = 1/|x|ln a. Where did that ln a come from? And why wasn't it there before? I shall answer both those questions... but in reverse.

#2 Why wasn't the ln a there before? The answer is it was! But why should they put ln of the base when the base is e?
Derivative of ln x= log (base e) x= 1/|x| ln e = 1/|x|*1 Does that make sense? The ln of an e simply becomes 1. So D/dx of ln x is just 1/|x|

#1 Ok well why put the ln a in there in the first place? Who came up with that! Well my explanation will be the easy one but we have to remember our logarithms. So as my student asked me about this problem it suddenly occurred to me to do some algebra so I made my Log(base a)x into a Ln x/ Ln a using a special rule called "Change of Base Rule". If you have never heard of the change of base rule stop reading about derivatives and integrals and open up your college algebra book. After doing this change of base rule it is easy to spot that ln a IS ONLY A CONSTANT! Once pulled out of the derivative we have (1/lna) d/dx ln x left. So the resulting 1/|x| is multiplied back with the constant we pulled out of the denominator just a moment ago. 1/|x|ln a is our easily derived derivative simply using the change of base rule. :)

P.S. All dx's dropped at the end of the derivatives were not missed. I simply didn't want to put them. Assume x's represent constants if it makes you feel better. aka d/dx ln x is 1/|x| dx not 1/|x|


Joshua C.


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