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Logarithm Formula

Logarithm are the inverse phenomena of exponential. Log function “undo” the exponential function. Consider a number x in the exponent equals to a fixed number b, the logarithmic value of b will be equal to the number x. Such as:

\[\large e^{x} = b\]

Taking log on both the sides, we have

\[\large log _{e}\; e^{x} = log_{e} \; b\]

\[\large x = log_{e} \; b\]

Note: It is to be noted that the logarithmic values of positive numbers is only known, i.e. x>0

Logarithmic value of 0- The logarithmic value of zero is undefined.

log(0) = Undefined

Logarithmic value of Negative number- The logarithmic value of negative numbers are imaginary, i.e. we don’t have a real value for negative numbers.

Below given are some important identities of the logarithmic function.

Trivial Identities

\[\large \log _{b} (1) = 0; \; because \; b^{0}=1; \; b> 0\]
\[\large \log _{b} (b) = 1; \; because \; b^{1}=b\]

Basic Logarithm Formulas

\[\large \log _{b} (xy) = \log _{b}(x) + \log _{b}(y)\]
\[\large \log _{b}\left ( \frac{x}{y} \right ) = \log _{b}(x) – \log _{b}(y)\]
\[\large \log_{b}(x^{d})= d \log_{b}(x)\]
\[\large \log_{b}(\sqrt[y]{x})= \frac{\log_{b}(x)}{y}\]
\[\large c\log_{b}(x)+d\log_{b}(y)= \log_{b}(x^{c}y^{d})\]

Changing the Base

\[\large \log_{b}a = \frac{\log_{d}(a)}{\log_{d}(b)}\]

Addition & Subtraction

\[\large \log_{b} (a+c) = \log_{b}a + \log_{b}\left ( 1 + \frac{c}{a} \right )\]
\[\large \log_{b} (a-c) = \log_{b}a + \log_{b}\left ( 1 – \frac{c}{a} \right )\]


\[\large x^{\frac{\log(\log(x))}{\log(x)}} \; = \; \log(x)\]

More topics in Logarithm Formula
Natural Log Formula Change of Base Formula
Exponential Growth Formula
Byjus Formulas