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5. Natural Logarithms (to the base e)

by M. Bourne

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Change of base

The number e frequently occurs in mathematics (especially calculus) and is an irrational constant (like π). Its value is e = 2.718 281 828 ...

Apart from logarithms to base 10 which we saw in the last section, we can also have logarithms to base e. These are called natural logarithms.

We usually write natural logarithms using `ln`, as follows:

`ln x` to mean `log_e x` (that is, "`log x` to the base `e`")

Natural logarithms are commonly used throughout science and engineering. (For example, see Applications of Derivatives of Logarithms.)

Where does this value "e" come from? Go to Calculating the Value of e to find out.


ln button on a calculator

NOTE: Please don't write natural log as

"In" (as in "She lives IN Singapore.")

Make sure it is

"LN - natural logarithm" (as in L for logarithm and N for natural).

I know it looks like "In" on your calculator because of the font they use, but you only confuse yourself if you don't write it properly.

Actually, the `ln` notation confuses a lot of students and it would be better if we (and calculators) wrote it our in full. That is `log_e`.

Example 1

Find the natural logarithm of `9.178`.


This means "Find `log_e 9.178`", which we can also write as "Find `ln 9.178`".

Using our calculator, we get

ln 9.178 = 2.2168

Check: Using the definition of a logarithm, we check as follows: `2.718\ 281\ 828 ^2.2168 = 9.1781`.

It checks OK.

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Continues below

Change of Base

At times we need to change from one base to another. The change of base formula (to change from base a to base b) is as follows:

`log_b x=(log_a x)/(log_a b)`

Example 2

Find the value of `log_3 8.7`.


[This problem is the same as answering: `3^?= 8.7`].

We cannot find `log_3 8.7` on a calculator. We need to use the change of base formula.

In this case, `b = 3` and `x = 8.7`.


`log_3 8.7=(log_10 8.7)/(log_10 3)` `=0.9395192/0.4771212` `=1.9691414`

CHECK: By calculator, `3^1.9691414 = 8.6999989`.

[We could have used natural logs as well, `log_3 8.7=(ln 8.7)/(ln 3)` which will give us the same answer.]


1. Use logarithms to base `10` to find `log_2 86`.


We estimate an answer in the range: `6` to `7`, because `2^6=64` and `2^7=128`, and `86` is between these 2 values.

`log_2 86 = (log 86)/(log 2)` `=1.934498451/0.301029995` `=6.426264755`

Our answer is between `6` and `7`, as expected.

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2. Find the natural logarithm of `1.394`.


`ln 1.394 = 0.332177312`

Check: This means `e^0.332177312=1.394`

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See also the Interactive Log Table where you can easily find log values to different bases.

Application of Exponential Functions

$$ dollar signs

Here is an exponential graph which made lots of people very RICH (as long as they sold out at the peak).

Check out the Dow Jones Industrial Average graph.


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