Chemistry, Third edition

2 1 · NUMBERS, UNITS AND MEASUREMENT

Now let us look at an example of standard notation. Think of the number 100. This

is the same as 1 100. In standard notation we write this as 1  102. Similarly,

2300 becomes2.3 1000 or2.3 103

6 749 008 becomes6.749 008  106

0.001 245 0 becomes1.2450 10 ^3

Logarithms

Logarithms to the base 10

The logarithm (or ‘log’) of a number to the base ten is the power that the number

10 has to be raised to in order to equal that number. For example, 100  102.

Therefore, the log of 100 is 2. Similarly, since 0.0001  1  10 ^4 , the log of 0.

is4.

What is the log of 150? The log of 150 is the value of xin the expression

150  10 x

We carry out this operation on a calculator. In many types of calculator, this is

done by entering log, the number, and then pressing the = button. The log of 150 is

2.176. We write this as

log (150) 2.

What if you are provided with the log of xand asked to find x? Using the above

example, how do we get back to 150 from 2.176? To do this we would need to evalu-

ate 102.176. (To carry out this operation on a scientific calculator we use the 10xkey,

a common sequence of operations being shift, 10 x,number, and =.) We then write

10 2.176 150

Similarly,

10 0.91040.

Logarithms to the base e (natural logs)

The symbol ‘e’ is a mathematical constant (like ) where

e2....

The logarithm of a number to the base e is the power that the number e has to be

raised to in order to equal that number. For example, e3.91250 so that the

Standard notation

Express the following in standard notation

(i) 0.000 0345 (iv)3.

(ii) 300 000 000 (v) 602 200 000 000 000 000 000 000

(iii)0.082 057 5 (vi) 17

Exercise 1A