Wednesday, December 30, 2015

logarithms - Why is ax=exloga?




Why is ax=exloga, where a is a constant?



From my research, I understand that the the natural log of a number is the constant you get when you differentiate the function of that number raised to the power x. For example, if we differentiate the function 2x, we get 2log2. I also understand that the natural log of e is just 1. But I cannot connect the dots here.



I would really appreciate an intuitive explanation of why we can write a number raised to a power as e raised to (the power x the natural logarithm of the number)?


Answer



I think we can agree that



a=eloga




which arises from one of the properties of the logarithm. Therefore, it’s sufficient to say that



ax=elogax



But one of the properties of the logarithm also dictates that



logax=xloga



Therefore




ax=exloga


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