Question

$\large{\underline{\underline{\mathrm{\red{Question-}}}}}$

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Write about logarithm and its rules. Also tell which base we use in physics and chemistry.

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Answer 1

$\huge{\underline{\rm{\red{Logarithm-}}}}$

Logarithm is the reverse function of exponentiation. Its basic use is to solve powers. It is all about exponents.

†Logarithm Function:

$\large{\boxed{\sf{\blue{f(x) = log_{b} (x)}}}}$

$\bold{Where}Where \begin{gathered}\begin{cases}\text{b and x are greater than zero}\\\text{b is not equal to 1}\end{cases}\end{gathered} </p><p>$

†Exponential Form examples:

$★ \large{\sf{log_{a} (x) = n}}$

$\implies⟹ \large{\sf{a^{n} = x}}$

$★ \large{\sf{log_{2} (16) = 4}}$

$\implies⟹ \large{\sf{2^{4} = 16}}$

$★ \large{\sf{log_{e} (y) = x}}$

$\implies⟹ \large{\sf{e^{x} = y}}$

$★ \large{\sf{log_{b} (a) = c}}$

$\implies⟹ \large{\sf{b^{c} = a}}$

†Bases that we generally use in chemistry and physics:

$\large{\sf{log_{e} (x)}}$

(x) , in this function, we have used base "e" which is equal to 2.7. This logarithm in which we use "e" as a base is called natural logarithm.

$\large{\sf{log_{10} (x)}}$

(x) , this function is mostly used in calculations.

†Conversion:

$\large{\boxed{\green{\sf{log_{e} x = 2.303 log_{10} x}}}}$

†RuLes:

$★ \large{\sf{log_{a} (mn) = log_{a} (m) + log_{a} (n)}}$

$★ \large{\sf{log_{e} ( \dfrac{m}{n} ) = log_{e} (m) - log_{e} (n)}}$

$★ \large{\sf{log_{e} (m^{n}) = n. log_{e} (m)}}$

$★ \large{\sf{log_{a} (a) = 1}}$

$★ \large{\sf{log_{a} (1) = 0}}$

†Base change rule:

$\implies⟹ \large{\sf{log_{b} (x) = \dfrac{log_{c} (x)}{log_{c} (b)}}}$