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 as:

ln x to mean logex (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

NOTE: Please don't write natural log as

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

Make sure it is

"math expression" (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.

Example 1

Find the natural logarithm of 9.178.

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:


math expression


Example 2

Find the value of log38.7

Let's have a look at this example in LiveMath. LiveMath works out the value of the expression exactly. In the second part, we compare using the change of base formula. We get the same result.

LIVEMath


Exercises

1. Use logarithms to base 10 to find log286.

2. Find the natural logarithm of 1.394

math expression

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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