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I want to know the difference between the exponential symbol $a^x$ and $e^x$ in mathematics symbols and please give me some examples for both of them.

I asked this question because of the derivative rules table below contain both exponential symbol $a^x$ and $e^x$ and I don't know when should I use one of them and when should I use the another one.

Derivative rules table:

enter image description here

[Derivative rules table source]

Mohammad Fakhrey
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4 Answers4

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The letter $e$ denotes this number, whereas the letter $a$ can be any positive real number.

Because $\ln(e)=1$ (essentially by definition), the rule $$\frac{d}{dx}(e^x)=e^x$$ is consistent with, and indeed a special case of, the rule $$\frac{d}{dx}(a^x)=a^x\ln(a)$$ There is no "choosing when to use one or use the other".

Zev Chonoles
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  • Thank you, but what about $x$ in $e^x$ .. any variable ?? – Mohammad Fakhrey Jul 07 '13 at 19:55
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    Any letter or any symbol is as good as any other (except for letters like $e$ that mathematicians have reserved for some special uses). It doesn't change the correctness of the statement to write $$\frac{d}{dt}(e^{t})=e^{t}, \qquad\frac{d}{d\clubsuit}(e^{\clubsuit})= e^{\clubsuit}, \qquad \frac{d}{d\phi}(e^{\phi})=e^{\phi}, \qquad \frac{d}{d{\it#}}(e^{\it#})=e^{\it#}$$ – Zev Chonoles Jul 07 '13 at 20:04
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    @MohammadFakhrey If you're asking what values $x$ can take, it can take any real number value, not just whole numbers. It would be impossible to take the derivative of a function where $x$ can only be a whole number. – Jack M Jul 07 '13 at 22:07
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You should think of the exponential function as being the primitive object. It is the inverse to the logarithm $\log:(0,\infty)\rightarrow\mathbf{R}$ (when I write $\log$ I mean the natural log, also denoted sometimes as $\ln$, not the base 10 logarithm). That is, $\log(e^x))=x$ and $e^{\log(y)}=y$ for $x\in\mathbf{R}$ and $y\in(0,\infty)$.

If you know what the exponential function is, then you can understand the function given by sending $x$ to $a^x$ for a positive real number $a$. By definition, $a^x$ means $e^{x\log(a)}$. Remember that here $\log(a)$ is just a number, the (natural) logarithm of $a$. Keeping this in mind, and remembering that the derivative of the exponential function is exist, you can use the chain rule to derive

$(d/dx)(a^x)=(d/dx)(e^{x\log(a)})=e^{x\log(a)}(d/dx)(x\log(a))=\log(a)e^{x\log(a)}=\log(a)a^x$.

Keenan Kidwell
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    How do you define log? I learn about these functions the other way around, exp being the solution of the differential equation $f' = f$ with $f(0) = 1$. – TonioElGringo Jul 08 '13 at 02:30
  • Dear @TonioElGringo, Then you could define the log as the inverse of the exponential function. – Keenan Kidwell Jul 08 '13 at 02:32
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    Yes, I know, but your answer seems to assume that the log function is defined and known. I wanted to know what definition you have for it that doesn't involve the exponential function. – TonioElGringo Jul 08 '13 at 02:33
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    What definition I have for the logarithm without the exponential? I guess you could define it as the integral $\int_1^t(1/x)dx$. But people see the exponential before they see integrals I think usually. The right definition, in my opinion, is as the inverse of the exponential. This is the definition that generalizes to complex analysis. – Keenan Kidwell Jul 08 '13 at 02:37
  • This is why I found it weird when you defined the exponential as the inverve of the natural logarithm in your answer. – TonioElGringo Jul 08 '13 at 02:40
  • I didn't define it that way. I just pointed out that it is the inverse of the log because it motivates the definition for the notation $a^x$ in the sense that, if $x$ is an positive integer, we get what we usually would mean by $a^n$. – Keenan Kidwell Jul 08 '13 at 02:40
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The two are essentially the same formula stated in different ways. They can be derived from each other as follows:

Note that $$\frac{d}{dx}(e^x)=e^x \ln(e) = e^x$$ is a special case of the formula for $a^x$ because $e$ has the special property that $\ln (e) =1$

Also $a^x=e^{\ln(a) x}$, which is another way into the derivative for $a^x$.

$$\frac{d}{dx}(e^{rx})=re^{rx}$$ by the chain rule. Let $r=\ln (a)$.

Mark Bennet
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The derivative of $e^x$ is $e^x\ln(e)$, so it's not different in that respect from other bases. But $\ln(e)$ is $1$.

That's what's "natural" about the number $e$.

Michael Hardy
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