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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 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 = 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.

 SolutionWrite 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 xloge 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: 