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Natural Logarithmic Function

Q: Why is e the base of the natural logarithm function?

Because f(x) = e x is the natural growth function, and the natural logarithm is the inverse of the natural growth function.

If you invest $1000 at 5% interest compounded continuously, how long will it take for you to be a millionaire? Here we will look at:

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Fact Checked Content
Last Updated: 01.06.2022
5 min reading time

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the definition of the natural logarithmic function and its relation to the natural exponential function,
how to graph the natural logarithmic function, and
how to convert a logarithmic function to a natural logarithmic function.

Natural Logarithmic Function Definition

Remember that e is the base used in the exponential growth and decay function $g (x) = e^{x}$ . For more details see Exponential Growth and Decay. In addition, you know that exponential functions and logarithms are inverses of each other, so the inverse of the exponential growth function is $f (x) = \log_{e} (x)$ . However, this natural logarithm gets used so much that it has a shorthand:

The natural logarithm function $f (x) = \log_{e} (x)$ is the inverse of the exponential function $g (x) = e^{x}$ , and it is written $f (x) = \log_{e} (x) = \ln (x)$ . This is read as "f of x is the natural log of x".

The graph below shows the natural log is the reflection of the exponential growth function over the line $y = x$ .

Natural Logarithmic Function inverse exponential Vaia The natural log and the exponential growth function | Vaia Originals

In intuitive terms, the exponential function tells you how much something has grown given an amount of time, and the natural log gives you the amount of time it takes to reach a certain amount of growth. You can think of it as

$e^{t i m e} = h o w m u c h g r o w t h \ln (h o w m u c h g r o w t h) = t i m e$

Suppose you have invested your money into chocolate, with an interest rate of 100% (because who doesn't want to buy chocolate), growing continuously. If you want to see 20 times your initial investment, how long do you need to wait?

Answer:

The natural logarithm gives you the amount of time. Since $\ln (20) \approx 3$ , you would only need to wait about 3 years to see 20 times your initial investment. That is the power of continuous compounding!

The Domain of the Natural Logarithmic Function

Properties of the natural logarithmic function

Because the natural logarithmic function is just a logarithm base e, it has the same properties as the regular logarithmic function.

Properties of the natural logarithmic function:

it is a logarithm with base e
there is no y intercept
the x intercept is at $(1, 0)$
the domain is $(0, \infty)$
the range is $(- \infty, \infty)$
$\ln (e^{x}) = x$
$e^{\ln (x)} = x$

Why is $\ln (e) = 1$ ?

Answer:

One reason is that the natural log and the exponential function are inverses of each other, so

$\ln (e) = \ln (e^{1}) = 1$

But the more intuitive reason is that the natural log tells you how long it takes to reach a certain amount of growth. So asking you to find $\ln (e)$ is the same as asking you to find the amount of time it takes to reach "e" growth. But from the exponential function you know that it takes 1 unit of time for the function $g (x) = e^{x}$ to reach the value "e", so $\ln (e) = 1$ .

Converting other logarithmic functions to natural logarithmic functions

It can be helpful to change the base of logarithmic functions to see how they compare to each other. To do this use the Proportion Rule for logarithms,

\log_{b} (x) = \frac{\log_{a} (x)}{\log_{a} (b)}

Since you want to convert $\log_{b} (x)$ to $\log_{e} (x)$ , use $a = e$ to get

$\log_{b} (x) = \frac{\log_{a} (x)}{\log_{a} (b)} = \frac{\log_{e} (x)}{\log_{e} (e)} = \frac{\ln (x)}{\ln (b)}$

So $f (x) = \log_{b} (x)$ is equivalent to $f (x) = \frac{\ln (x)}{\ln (b)}$ .

Convert the functions $h (x) = \log_{2} (x)$ and $g (x) = \log (x)$ to base $e$ , then graph them all in the same picture.

Answer:

Remember that when a base isn't mentioned that it is assumed to be base 10. So using the Proportion Rule you get

$h (x) = \log_{2} (x) = \frac{\ln (x)}{\ln (2)},$

and

$g (x) = \log (x) = \frac{\ln (x)}{\ln (10)}$

So they are just constant multiples of the natural logarithmic function.

Comparing the natural log, log base 2, and log base 10 | Vaia Originals

Derivatives of the Natural Logarithmic Function

The derivative of the natural logarithmic function is

$\frac{d}{d x} \ln (x) = \frac{1}{x}$

For more information on the derivative of the natural logarithmic function see Derivative of the Logarithmic Function.

Integration of Natural Logarithmic Functions

The integral of the natural logarithmic function is

$\int \ln (x) d x = x \cdot \ln (x) - x + C$

For more information on the integral of the natural logarithmic function see Integrals Involving Logarithmic Functions.

Natural Logarithmic Function - Key takeaways

The natural logarithm and the exponential function are inverses of each other
The natural log of x is the amount of time it takes for the function $y = e^{x}$ to reach y amount of growth.
The natural logarithm function $f (x) = \log_{e} (x)$ is the inverse of the exponential function $g (x) = e^{x}$ , and it is written $f (x) = \log_{e} (x) = \ln (x) .$

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Frequently Asked Questions about Natural Logarithmic Function

What is the natural logarithm function?

The natural logarithm is a logarithmic function with a base of e, where e is Euler's number.

How to graph natural logarithmic functions?

The most intuitive way to graph the natural log function is to think of it as the inverse of the exponential function.

Why is e the base of the natural logarithm function?

Because f(x) = e^x is the natural growth function, and the natural logarithm is the inverse of the natural growth function.

How do you solve natural logarithmic functions?

You don't solve natural logarithmic functions, you solve natural logarithmic equations.

How to convert logarithmic function to natural logarithmic function?

Use the Proportion Rule for logarithms.

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