Unit 4: Mathematics Introduce the laws of Logarithms. Aims Objectives Identify the 4 laws of Logarithms Use the laws of Logarithms to calculate given formulas.

The laws of logarithms

The laws of logarithms Logarithms come in the form We say this as "log of to base ". But what does mean? means "What power of 5 gives 25?“ The answer is 2 because , = 25 in other words 25 = 2

The laws of logarithms means "What power of 2 gives 16 ?“ The answer is 4 because , = 16 , in other words So means "What power of a gives x ,?" Note that both a and x must be positive.

On the calculator we press- log (64)÷log (8) = 2 If we write down that 64 = then the equivalent statement using logarithms is On the calculator we press- log (64)÷log (8) = 2 a) 2 b) 5 c) 3 d) 4 e) 3 f) 1 2 g) 1 3 h) 1 4

The laws of logarithms Indices can be applied to any base. The tables of logarithms most useful in computations use a base of 10. These are called Common Logarithms. Any base could be used in theory. Base 10 simplifies the work involved in calculations because our number system is base 10. We can apply the laws of indices as before to base 10. 103 × 104 = 1,000 × 10,000 = 10(3 + 4) = 107 = 10,000,000

The laws of logarithms The logarithm of 1000 to base 10 is 3 (remember 103 = 1000). This is written: log101000 = 34 Because base 10 is so important, it is assumed if no base is indicated. The above can also be written simply as log (1000) = 3.

The laws of logarithms Note that the indices 3 and 4 (above) tell us how many zeros the numbers 1,000 and 10,000 contain. Here is a list of some whole number base 10 logarithms. Number Equivalent Logarithm 10,000,000 107 7 1,000,000 106 6 100,000 105 5 10,000 104 4 1,000 103 3 100 102 2 10 101 1

The laws of logarithms Note that the logarithm of 1 is 0. This is because 100 = 1. This makes sense. When you multiply a number by 1 you do not change its value. Correspondingly, if you add 0 to the index you leave it unchanged. 10 × 1 = 101 × 100 = 10(1 + 0) = 101 = 10

The laws of logarithms There is more (much more) to logarithms than the whole number values discussed so far. A number like 63 will have as its logarithm a number between 1 and 2. In fact, 63 can be written as 101.799340549.... so log(63) = 1.799340549....

The laws of logarithms Imagine that we wish to multiply two numbers, say, 63 and 41. By using tables of logarithms the two numbers can be written as 101.7993 × 101.6128 (to four decimals) The multiplication can then be done by adding the indices: 10(1.7993 + 1.6128) = 103.4121

The laws of logarithms Example 1 : If = 2 then Example 2 : We have 25 = . Then = 2.

1. (a) x = 3to9 (b) 8 = 2x (c) 27 = 3x (d) x = 43 (e) y = 25 (f) y = 52 (a) 4 = log3 y (b) x = log3 27 (c) 2 = log4m (d) 5 = log3 y (e) 5 = logx 32 (f) x = log4 64

The laws of logarithms Example 3 : If Then: Example 4 : If = 4 then

3. (a) 81 (b) 3 (c) 10 (d) 64 (e) 243 (f) 8

Properties of Logs Logs have some very useful properties which follow from their definition and the equivalence of the logarithmic form and exponential form. Some useful properties are as follows:

Properties of Logs

Example 3

Section 2 1. (a) log2 y/x (b) 4 + 3 log2 x (c) log3 y - 1 (d) 2 log4(xy) (e) 0

The Natural Logarithm and Exponential The natural logarithm is often written as ln which you may have noticed on your calculator. The symbol e symbolizes a special mathematical constant. It has importance in growth and decay problems. The logarithmic properties listed above hold for all bases of logs. If you see log x written (with no base), the natural log is implied. The number e can not be written

The Natural Logarithm and Exponential exactly in decimal form, but it is approximately 2:718. Of course, all the properties of logs that we have written down also apply to the natural log. In particular, are equivalent statements. We also have . = 1 and ln 1 = 0.

1. (a) 0.34 (b) -0:14 (c) 0.89 (d) 1.86 (e) 44.70 (f) 1.62 (g) 9.03 (h) 0.10 (i) 3.41 (j) 4.65

The first law of logarithms Suppose then the equivalent logarithmic forms are Using the first rule of indices Now the logarithmic form of the statement

The first law of logarithms But and and so putting these results together we have So, if we want to multiply two numbers together and find the logarithm of the result, we can do this by adding together the logarithms of the two numbers. This is the first law.

The second law of logarithms Suppose x = or equivalently = n. Suppose we raise both sides of x = to the power m: Using the rules of indices we can write this as Thinking of the quantity as a single term, the logarithmic form is

The second law of logarithms Suppose x = or equivalently = n. Suppose we raise both sides of x = to the power m: Using the rules of indices we can write this as Thinking of the quantity as a single term, the logarithmic form is

The third law of logarithms As before, suppose and with equivalent logarithmic forms a and (2)

The third law of logarithms using the rules of indices. In logarithmic form which from (2) can be written

The logarithm of 1 Recall that any number raised to the power zero is 1: The logarithmic form of this is

Example 10 = 3 x 5 = 15 or log (15) ÷ log (10)= 1.176091259 Therefore we can write this as: 1.176091259 10