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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 loge.

For example:

loge 57 = ln 57

loge 4.92 = ln 4.92

loge 0.56 = ln 0.56

Note:

The abbreviation “ln” comes from the French term for natural log, “logarithme népérien.”

 

Definition — Natural Logarithm

A natural log is a log whose base is e. It is written like this:

ln x = logex

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 e3

Solution

The base of a natural log is e.

Therefore, write ln e3 as logee3.

Set x equal to loge e3.

Rewrite in exponential form.

Use the Principle of Exponential Equality to solve for x.

So, ln e3 = 3.

 

 

x =

ex =

x =

 

logee3

logee3

e3

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 loge x.

Rewrite in exponential form.

Use your calculator to compute e7.92.

Then round to the nearest whole number.

 So, ln 2752 7.92.

ln x

loge x

e7.92

 

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

loge 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:

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