1. The Natural logarithm and e
IB SL/HL
Adrian Sparrow

2. Learning outcome:
you should be able to solve equations by using the relationship between the natural logarithm and e.

3. Natural logarithms
A logarithm to the base e is called a natural logarithm.
loge x is written as ln x
Point out that the domain of y = ln x is the set of positive real numbers. In other words, ln x is not defined for negative values of x (or x = 0). This follows from the fact that ex is always positive.
The range of ln x, on the other hand, is the set of all real numbers.
The graph of y = ln x is asymptotic to the y-axis.
So,
ln x is the inverse function of ex

4. Using the natural log - ln
Without using a calculator find the value of:
Use a calculator to find:
= 3 =0
= 4 =1 = =2
= -3 =
= -1
= n

5. The laws of natural logarithms
It is also helpful to realise that
ln ex = x

6. Finding a missing index using logarithms
Find x to 2 decimal places using trial and error.
This process can be used with any base log, even the natural log.
Far too complicated ...
Take a log of both sides
Now try these, answers to 2 d.p.
Use the power rule

7. Remember the base of a natural log is e.
Find x, if
Find x, if
Remember the base of a natural log is e.
Take a natural log of both sides.
Rearrange in index form.
Use the power rule.
Find x in each of the following:
Find x in each of the following: