The natural logarithm function, ln x , is the inverse of the exponential function e^x .

You will often have to use the natural logarithm function when solving problems about exponential growth and decay. You will also meet the natural logarithm function in your calculus module, where it is closely related to the area bounded by certain curves.

There are three rules which help us to simplify mathematical expressions which contain logarithms. By the end of this activity, you should have found out what they are.

Simplify the following:

1) ln(5) + ln(2)         Ans:

2) ln(7) + ln(3)       Ans:

3) ln(2) + ln(11)       Ans:

4) ln(5) + ln(3) + ln(2)       Ans:

5) ln(10) + ln(7) + ln(11)       Ans:

6) ln(1) + ln(2) + ln(43)       Ans:

Simplify the following:

7) ln(12) - ln(4)       Ans:

8) ln(99) - ln(3)       Ans:

9) ln(49) - ln(7)       Ans:

10) ln(100) - ln(20)       Ans:

11) ln(12) - ln(8)       Ans:

12) ln(13) - ln(1)       Ans:

Simplify the following:

13) ln(25)       Ans:

14) ln(36)       Ans:

15) ln(100)       Ans:

16) ln(8)       Ans:

17) ln(27)       Ans:

18) ln(16)       Ans:

19) ln(256)       Ans:

20) ln(125)       Ans:

Simplify the following:

1) ln(    ) + ln(    )         Ans: ln(35)

2) ln(    ) + ln(    )         Ans: ln(86)

3) ln(    ) + ln(    ) + ln(    )         Ans: ln(70)

4) ln(    ) + ln(    ) + ln(    )         Ans: ln(42)

5) ln(    ) - ln(    )         Ans: ln(21)

6) ln(    ) - ln(    )         Ans: ln(2)

7) ln(    ) - ln(   )         Ans: ln(3)

8) ln(    )         Ans: 2 ln(7)

9) ln(    )         Ans: 2 ln(9)

10) ln(    )         Ans: 3 ln(7)

11) ln(    )         Ans: 4 ln(3)

12) ln(    )         Ans: 5 ln(10)

Write out in your own words what you think the three rules of logarithms are.

Before you start, make sure you are in "exact" mode by pressing:

<O>ptions

<P>recision

<E>xact <enter>

Strictly speaking, it is not necessary to include the brackets.

For example, ln(5) could just be written as ln 5 .

However, DERIVE prefers to use brackets in its output. Later, in more complicated work, it is essential to use brackets. So get into the habit of using brackets now.

To answer these questions, simply do the following:

<A>uthor . . . the given expression with logarithms . . .

<S>implify

After the first few, can you predict what might happen for the rest?

Based on the patterns you noticed in Exercise 1, guess what numbers you think should go in the brackets. The Author and Simplify the your question, and check you get the given answer.

IMPORTANT NOTE: There is more than one correct way of doing these. Compare your results with your classmates.

David Bowers,
Mathematics Workshop,
Suffolk College.