Natural Logs
Notation for Natural Logs
A natural logarithm is a log with base e. The number e is an irrational
number whose value is approximately 2.718.
Natural logs are usually written with the abbreviation ln rather than log_{e}.
For example:
log_{e }57 = ln 57
log_{e }4.92 = ln 4.92
log_{e} 0.56 = ln 0.56
Note:The abbreviation â€œlnâ€ comes from the
French term for natural log,
â€œlogarithme nÃ©pÃ©rien.â€
Definition â€”
Natural Logarithm
A natural log is a log whose base is e. It is written like this:
ln x = log_{e}x
Here, x > 0.
Note:
The number e occurs naturally in
applications such as compound interest,
population growth, and radioactive decay.
Finding Natural Logs
We can find the value of some natural logs by switching from logarithmic
notation to exponential notation.
Example 1
Find: ln e^{3}
Solution
The base of a natural log is e.
Therefore, write ln e^{3} as log_{e}e^{3}.
Set x equal to loge e^{3}.
Rewrite in exponential form.
Use the Principle of Exponential Equality to solve for x.
So, ln e^{3} = 3. 
x =
e^{x} =
x = 
log_{e}e^{3}
log_{e}e^{3}
e^{3}
3 
In the last example we did not need a calculator because the argument was
a power of e. However, in many cases we will need a calculator to find the
natural log of a number.
Example 2
Solve: ln x = 7.92. Round to the nearest whole number.
Solution Write ln x as log_{e
}x.
Rewrite in exponential form.
Use your calculator to compute e^{7.92}. Then round to the nearest whole number. So, ln 2752
≈ 7.92. 
ln x
log_{e} x
e^{7.92}
2752 
= 7.92 = 7.92
= x
≈ x 
Example 3
Solve: ln x = 2.53
Solution
Write ln x as logex.
Rewrite in exponential form.
Use your calculator to compute e^{2.53}.
Then round to two decimal places.
So, ln 0.08 ≈ 2.53. 
ln x log_{e} x
e^{2.53}
0.08 
= 2.53 = 2.53
= x
≈ x 
Note:
Remember, the number e^{2.53} is a positive
number, even though the exponent is
negative.
To show this, we can write:
