Lesson 9, Topic 1
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# Definition of Logarithm

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Logarithm of a number to a given base is the index to which the number must be raised to give the number.

Expressed mathematically;

$$\scriptsize b^x = a \: \leftrightharpoons \: \log_b a = x$$

Here are examples to help you understand the conversions from one form to the other form.

a. $$\scriptsize 3^2 = 9 \: \leftrightharpoons \: \log_3 9 = 2$$

b. $$\scriptsize 4^3 = 81 \: \leftrightharpoons \: \log_4 81 = 3$$

c. $$\scriptsize 8^y = 2 \: \leftrightharpoons \: \log_8 2 = y$$

Other examples:

### Common Logarithms:

Common logarithm is simply, a log with base 10. Base 10 is used because it is the base of our decimal system of numbers. A common log means log10. But usually, writing “log” is sufficient instead of writing log10. i.e., log100 = log10100.

So when you see a log with no base it means that it is log10.

Examples:

a. $$\scriptsize 10^2 = 100 \: \leftrightharpoons \: \log_{10} 100 \: or \: \log 100 = 2$$

a. $$\scriptsize 10^{-2} = 0.01 \: \leftrightharpoons \: \log_{10} 0.01\: or \: \log 0.01 = -2$$

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