01Ylogintro

= Introduction to Logarithms =

We now understand that: ... ... 2 0 = 1 ... ... 2 1 = 2  ... ... 2 2 = 4  ... ... 2 3 = 8  ... ... etc

We also leaned about fractional indices math \\ .\qquad 2^{\frac{1}{2}} = \sqrt{2} \\. \\ . \qquad 2^{\frac{1}{3}} = \sqrt[3]{2} math

We could write this using decimals: ... ... 2 0.5 = 1.4142 {approximately} ... ... 2 0.33 = 1.2600 {approximately}

So, putting these together (notice the increasing values): ... ... 2 0 = 1 ... ... 2 0.33 = 1.2600 {approximately} ... ... 2 0.5 = 1.4142 {approximately} ... ... 2 1 = 2 ... ... 2 2 = 4

This raises the question: What about other decimal indices? ... ... 2 0.1 = ? ... ... 2 0.2 = ?  ... ... 2 1.6 = ?  ... ... 2 2.8 = ?

Your calculator will give an answer to these (use the y x button): ... ... 2 0.1 = 1.0718 ... ... 2 0.2 = 1.1487  ... ... 2 1.6 = 3.0314  ... ... 2 2.8 = 6.9644


 * BUT **
 * We know 2 3 = 2 × 2 × 2
 * What does 2 2.8 mean??

Ultimately, there is no simple answer to that question. The best answer can be found by plotting a graph of x against y = 2 x : and joining the points with a smooth curve.



From the graph we can see that 2 2.8 = 6.96 is a point on the smooth curve joining the known powers of 2.

This raises the question:
 * What index (with a base of 2) would give 7 as the answer?
 * 2 a = 7 ... a = ?

This is a common question in maths We use the word ** logarithm ** (or ** log **) to describe the unknown index.
 * If the base is 2, write a small subscript 2 after the word "log"
 * The same question can therefore be written as
 * a = log 2 (7)

We could use trial and error on your calculator (with the y x button) to find an approximate answer:
 * How do we calculate log 2 (7)? **
 * 2 2.8 = 6.9644
 * 2 2.81 = 7.0128
 * 2 2.809 = 7.0080
 * 2 2.808 = 7.0031
 * 2 2.807 = 6.9983
 * 2 2.8071 = 6.9988
 * etc

Or we could use graphing software (such as Graphmatica) and find the intersection between
 * y = 2 x ... (and)
 * y = 7


 * Note: **
 * There is nothing special about using 2 as a base
 * We could use base 3:
 * 3 a = 12
 * a = log 3 (12)
 * We could use base 5
 * 5 a = 31
 * a = log 5 (31)
 * We could use base 10
 * 10 a = 74
 * a = log 10 (74)


 * Calculating Logs on a scientific calculator **

Your calculator has a button labelled: **log ** This calculates log with a __**base of 10**__

To find log 10 (74)
 * Example: **
 * Type: **log ** 74 **= **
 * Answer: 1.8692
 * So: 10 1.8692 = 74

To find a such that 10 a = 123
 * Example: **
 * a = log 10 (123)
 * Type: **log ** 123 **= **
 * Answer: 2.0899
 * So: 10 2.0899 = 123

To find logs with other bases using a scientific calculator is a tiny bit more tricky. You have to use the change of base rule:

math . \qquad \log_2{7} = \dfrac{ \log_{10}{7} }{ \log_{10}{2} } math

To find a such that 2 a = 7
 * Example: **
 * a = log 2 (7)
 * type: **log ** 7 **÷ log ** 2 **= **
 * Answer: 2.8074

To find a such that 5 a = 31
 * Example: **
 * a = log 5 (31)
 * type: **log ** 31 **÷ log ** 5 **= **
 * Answer: 2.1337


 * Applications of Logarithms **

Logarithms are used in a variety of ways:
 * measuring the loudness of sounds (decibels)
 * measuring the strength of earthquakes (Richter Scale)
 * Designing slide rules

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