logarithms - Why is $ a^x = e^{x log a} $?

Why is $ a^x = e^{x \log a}$, 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 $ 2^x $, we get $ 2 \log 2$. 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

$$a=e^{\log a}$$

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

$$a^x=e^{\log a^x}$$

But one of the properties of the logarithm also dictates that

$$\log a^x=x\log a$$

Therefore

$$a^x=e^{x\log a}$$