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The information on page 21 (Indices and logarithms) can be divided into a number of distinct sections as shown by the red boxes above. Indices and logarithms are different ways of looking at the same thing. The key to the whole thing is the expression in the section in the box called “definition” above.

The equation on the left of this expression is in exponential form. The equation on the right is in logarithmic form.

The symbol <=> is read as “if and only if ”.

Base ^{Index} = value <=> log_{base} value = Index

For example, in the statement: 2^{3} = 8

2 is called the **base**,

3 is called the **index **or** exponent**,

2^{3} is called the **power**,

8 is called the **value** of the power.

The above example is read as 2 raised to the power of 3 is 8.

Note that the power consists of two parts; a base and an index. To be correct about it, index is not another word for power, but in practice the word power is often incorrectly used instead of index. Indices is the pleural of index.

These **laws of indices** give the rules for dividing and multiplying numbers written in index form.

These properties only hold when the same number is being raised to a certain power. For example, we cannot easily work out what 2³×5² is, whereas we can simplify 3²×3³

Log is used as a short form of logarithm, and is another name for index. All logs must be in the same base in applying the rules and solving for values. If this is not the case at the start of a problem to be solved then use the law of logs in the red box labelled “Change base” in the picture above. There are two bases commonly used. If working with base 10 it can be written as log_{10} or log. If no base is specified it is assumed to be base 10 as this is the base we normally work in in our everyday life. The other common base is the base e, (where e is approximately 2.718281…). Logs to the base e are called Natural Logarithms and are written as log_{e}x or ln x.

Note the similarity between the laws of indices on the left and the corresponding laws of logs connected above by the blue lines.

**Antilogs**

We also need to know about **antilogs**. The antilogarithm is the number for which a given logarithm stands; for example, where log x equals y, then x is the *antilogarithm* of y. For example, the *antilogarithm* of 3 to the base 10 is 1,000, or antilog 3 = 1,000, or log 1,000 = 3.

When we work with numbers in index form we generally prefer the index to be a whole number. We don’t like things like x = 10^{16.84719}. To put this in a more acceptable format we enter the index 16.84719 into our calculator and use the second function or shift key and then antilog to get an equivalent value for x with the 16.84719 changed to a whole number. Of course if the index changes then the base number (10 in our example) must also change for the overall value of x to remain the same. When we do this the calculator returns 7.03 x 10^{16} to three decimal places. 10^{16.84719} is the same as 7.03 x 10^{16}.

**Question.**

Solve for x if log_{8 }x – log_{9 }x = 1

**Answer**

log_{10 }x log_{10 }x

——– – ——– = 1 Bring to the same base (base 10).

log_{10 }8 log_{10 }9

When working with base 10 do not have to write the “10”. If no base is given, then base 10 is assumed.

In this example remove fractions by multiplying across by denominators.

log x . log 9 – log x . log 8 = log 8 . log 9

log x ( log 9 – log 8) = log 8 . log 9

Using calculator

log x ( 0.954242509 – 0.90308998) = 0.954242509 X 0.90308998

0.051152528 log x = 0.8617765913

No need for too many decimal places, but because we are using an approximate value instead of an exact value we use the curvy approximately ~ sign instead of the = equals to sign.

log x = 0.8617765913 / 0.051152528 = 16.84719

x ~ 10^{16.84719}

We usually like to have our index as a whole number, 16.84719 is not.

So we use antilogs to convert 10^{16.84719} to an equivalent value that has the index given as a whole number. Using calculator find that antilog of 16.84719 (assuming 10 is raised by this index) is 7.03 x 10^{16}.

Answer: x ~ 7.03 x 10^{16}.

Graph of Log functions

The number e and the natural log.

Properties of logs