# Why do we use the natural log in the derivative of an exponential function?

• QuickLoris
In summary: When we use the natural log, we are saying that we are going to base the log on the natural base, which is e. So, in other words, since e is defined so that lim e^{h}=1 as h\rightarrow0, the derivative is itself. Otherwise, the derivative would be recursive? as in, f(x) = a^x \frac{d}{dx}f(x) = a^{x}\frac{d}{dx}f(0)
QuickLoris
I recently struck a question that I have not been able to find an answer to. I feel like I'm missing something obvious, so I've come here for help.

The derivative of $a^{x}$ is $a^{x}$lna.

The explanation that Stewart 5e gives is:
$\frac{d}{dx}$$a^{x}$ = $\frac{d}{dx}$$e^{(lna)x}$

= $e^{(lna)x}$$\frac{d}{dx}$(lna)x

=$e^{(lna)x}$$\cdot$lna

=$a^{x}$lna

My question is: Why do we use the natural log instead of a log of any other base?

ax = e(lna)x

d/dx(ecx) = cecx

If you used any base other than e, the second equation would be a problem.

So, in other words, since e is defined so that lim $e^{h}$=1 as h$\rightarrow$0, the derivative is itself. Otherwise, the derivative would be recursive? as in,

f(x) = $a^{x}$

$\frac{d}{dx}$f(x) = $a^{x}$$\frac{d}{dx}$f(0)

Is that right?

Hey QuickLoris and welcome to the forums.

The natural base has so many properties for so many applications including pure mathematics, applied mathematics, and statistics, that it is just well suited for these things and as such it becomes not only a tool of frequent use, but also one of investigation.

You have for example the connection between the trig functions to the hyperbolic ones and the exponential via Eulers formula and the complex valued analogs for the trig and hyperbolic.

In statistics you have probability transform functions, distributions, and a variety of other things involving the exponential function.

There are just so many connections that it becomes kind of a "neat coincidence" for all of mathematics.

Use any base you like

$$\dfrac{d}{dx}a^x=\frac{\log_b(a)}{\log_b(e)} a^x$$

We can see if b=a or e, we will only need one log.

QuickLoris said:
So, in other words, since e is defined so that lim $e^{h}$=1 as h$\rightarrow$0, the derivative is itself. Otherwise, the derivative would be recursive? as in,

f(x) = $a^{x}$

$\frac{d}{dx}$f(x) = $a^{x}$$\frac{d}{dx}$f(0)

Is that right?
No, it is not right. Since f(0) is a number, a constant, and does not depend on x, "df(0)/dx" is equal to 0.

lurflurf said:
Use any base you like

$$\dfrac{d}{dx}a^x=\frac{\log_b(a)}{\log_b(e)} a^x$$

We can see if b=a or e, we will only need one log.

I understand that you can use the change of base formula to to change the base to whatever you like once you have the derivative, I just wanted to know why ln was chosen to begin with. mathman somewhat answered my question.

HallsofIvy said:
No, it is not right. Since f(0) is a number, a constant, and does not depend on x, "df(0)/dx" is equal to 0.

I should have used different notation. I mean f '(0), not f(0).

## 1. Why do we use the natural log in the derivative of an exponential function?

The natural log, or ln, is used in the derivative of an exponential function because it is the inverse of the exponential function. This means that it helps us find the original exponent when we take the derivative of an exponential function. In other words, it helps us find the rate of change of the exponential function.

## 2. Can't we just use any other base for the logarithm in the derivative of an exponential function?

While we can use other bases for the logarithm, the natural log is the most commonly used in calculus because it has unique properties that make it easier to work with. It is also closely related to the exponential function, making it the most convenient choice for finding the derivative.

## 3. How does the natural log relate to the derivative of an exponential function?

The natural log is the inverse function of the exponential function, meaning that they "undo" each other. This relationship is why the natural log is used in finding the derivative of an exponential function. It allows us to find the rate of change of the exponential function by "undoing" the exponent.

## 4. Can we use the natural log in the derivative of any exponential function?

Yes, the natural log can be used in finding the derivative of any exponential function. This is because the natural log is a general logarithm that can be used to find the derivative of any exponential function, regardless of the base.

## 5. Is there any other reason we use the natural log in the derivative of an exponential function?

Another reason for using the natural log in the derivative of an exponential function is that it simplifies the process of finding the derivative. The natural log has properties that make it easier to differentiate, making it a more efficient choice for finding the derivative of exponential functions.

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