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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