**Math Logarithm Properties**Advertisement

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If **a** is positive real number, other than 1 and c^{n} = m, then we say: n = logc^{m} and we say that the value of logm to the base c is n.

- 10
^{3}1000 = log_{10}1000 = 3. - 3
^{4}= 81 = log_{3}81 = 4 - (.1)
^{2}= .01 = log_{(.1)}.01 = 2

**Logarithm Properties :**

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**Product Rule :**log_{a}(xy) = log_{a}x + log_{a}y**Quotient Rule :**log_{a}(x/y) = log_{a}x – log_{a}y**Logarithm of any quantity same base is unity**i.e, log x_{X}= 1**Logarithm of 1 to any base Zero**i.e, log_{a}1 = 0- log
_{a}(x^{n}) = n(log_{a}^{x}) - log
_{a}^{x}= 1 / log_{x}a **Change of Base Rule :**log_{a}^{x}= log_{b}^{x}/ log_{b}^{a}= log x / log a

- log
_{b}N = log_{b}a . log_{a}N, ( a > 0, a ≠ 1, N>0 )

- log
_{b}a = 1 / log_{a}b , (a > 0, a ≠ 1)

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**log**_{b}1 = 0**log**_{a}a = 1**log**_{b}0 = { – ∞, b > 1, + ∞, b < 1 }

**Decimal Logarithm**

- log
_{10}N = lgN ( b = 10)

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- lgN = x ⇔ 10
^{x}= N

**Natural Logarithm**

log_{e} N = InN

InN = x ⇔ e^{x} = N

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