**Understanding the Log Function.**

In the mathematical operation of addition we take two numbers and join them to create a third:

1 + 1 = 2. | (21) |

We can repeat this operation:

1 + 1 + 1 = 3. | (22) |

Multiplication is the mathematical operation that extends this:

(23) |

In the same way, we can repeat multiplication:

(24) |

and

(25) |

The extension of multiplication is exponentiation:

(26) |

and

(27) |

This is read ``two raised to the third is eight''. Because exponentiation simply counts the number of multiplications, the exponents add:

The number `2' is called the base of the exponentiation. If we raise an exponent to another exponent, the values multiply:

The exponential function *y* = 2^{x} is shown in this
graph^{4}:

Now consider that we have a number and we want to know how many 2's must be
multiplied together to get that number. For example, given that we are using `2'
as the base, how many 2's must be multiplied together to get 32? That is, we
want to solve this equation:

2^{B} = 32. |
(30) |

Of course, 2

(31) |

We pronounce this as ``the logarithm to the base 2 of 32 is 5''. It is the ``inverse function'' for exponentiation:

and

The logarithmic function is shown in this graph^{5}:

This graph was created by switching the *x* and *y* of the
exponential graph, which is the same as flipping the curve over on a 45 line. Notice in particular that and .

**The Addition Law.** Consider this equation:

(34) |

which is just a generalization of equation (28). Take the logarithm of both sides:

(35) |

Exponentiation and the logarithm are inverse operations, so we can collapse the left side:

(36) |

Now let's be tricky and substitute: and :

(37) |

Again, exponentiation and the logarithm are inverse operations, so we can collapse the two cases on the right side:

(38) |

This is the additive property that Shannon was interested in.

**The ``Pull Forward'' Rule.** From equation
(32):

(39) |

Raise both sides to the

(40) |

Now, we can combine the exponents by multiplying, as in (29):

(41) |

Finally, take the log base 2 of both sides and collapse the right side:

(42) |

This can be remembered as a rule that allows one to ``pull'' the exponent forward from inside the logarithm.

**How to Convert Between Different Bases.**
Calculators and computers generally don't calculate the logarithm to the base 2,
but we can use a trick to make this easy. Start by letting:

Rearrange it as:

(44) |

Now use a ``reverse pull forward'' (!):

(45) |

and drop the logs:

a = b^{x}. |
(46) |

Now take the log base

(47) |

This simplifies to:

(48) |

But we know what

(49) |

The conversion rule to get logarithms base 2 from any base z is:

(50) |

Notice that since the

(51) |

You should get `5'.

**Tricks With Powers of 2.** In calculus we
learn about the natural logarithm with base ^{6} Calculations with this base can easily be done
by a computer or calculator, but they are difficult for most people to do in
their head.

In contrast, the powers of 2 are easy to memorize and remember:

choices | bits |

M | B |

1 | 0 |

2 | 1 |

4 | 2 |

8 | 3 |

16 | 4 |

32 | 5 |

64 | 6 |

128 | 7 |

256 | 8 |

512 | 9 |

1024 | 10 |

We can use this table and a trick to make quick estimates of the logs of
higher numbers. Notice that

(52) |

So to take the log base 2 of , we think:

= | (53) | ||

= | (54) | ||

= | (55) | ||

(56) | |||

2 + 10 + 10 | (57) | ||

22 | (58) |

The actual value is 21.93.

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