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:

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

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

†Exponential Form examples:

★ $\large{\sf{log_{a} (x) = n}}$

$\implies$ $\large{\sf{a^{n} = x}}$

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★ $\large{\sf{log_{2} (16) = 4}}$

$\implies$ $\large{\sf{2^{4} = 16}}$

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★ $\large{\sf{log_{e} (y) = x}}$

$\implies$ $\large{\sf{e^{x} = y}}$

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★ $\large{\sf{log_{b} (a) = c}}$

$\implies$ $\large{\sf{b^{c} = a}}$

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†Bases that we generally use in chemistry and physics:

• $\large{\sf{log_{e} (x)}}$, 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.

• $\large{\sf{log_{10} (x)}}$, this function is mostly used in calculations.

†Conversion:

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

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†RuLes:

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

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

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

★ $\large{\sf{log_{a} (a) = 1}}$

★ $\large{\sf{log_{a} (1) = 0}}$

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†Base change rule:

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

Answer 2

Logarithm

Logarithm :

The Logarithm function is defined as

$\sf\:f(x) =\log_{b}(x)$

where b > 0 and b ≠ 1 and also x >0, reads as log base b of x.

⇒Generally we use the base 10 i.e $\log_{10}$

⇒if $\sf\:\log_{b}(a) = x$ ,in exponent form :

$\implies \: b {}^{x} = a$

•Types of Logarithm :

1. Natural or Napier Logarithms

The logarithms with base ‘e’are called natural logarithms. ( e = 2.71..)

e.g., loge x, loge 56 etc

2. Common Logarithms

The logarithm with base ’10’ are called common logarithm.

e.g., log10 x, log10 75 etc

Rules of Logarithm :

• Basic rules

$\sf\:1)\log(a) + \log(b) = \log(ab)$

$\sf\:2)\log( \frac{a}{b} ) = \log(a) - \log(b)$

$\sf\:3)\log(a) {}^{n} = nlog(a)$

$\sf\:4)\log_{a}(a) = 1$

$\sf\:5)\log_{a}(1)=0$

$\sf\:6)\log_{a{}^{n}}x=\dfrac{1}{n}\times\log_{a}x$

$\sf\:7)\log_{a{}^{n}}x{}^{m}=\dfrac{m}{n}\times\log_{a}$

•Base changing rule

$\sf\:\log_{a}x = \dfrac{\log_{b}x}{\log_{b}a}$

•More rules :

1)$\sf\:x{}^{\log_{a}(y)}=y{}^{\log_{a}(x)}$ , where x>0 , y > 0 ,a>0 and a≠ 1

2) $\sf\:\log_{x}(a)=\dfrac{1}{\log_{x}(a)}$ ,for a>0 ,a ≠ 1 and x> 0 and ,x≠ 1

Note :

• Logarithm function is commonly used in physical chemistry eg : In Electrochemistry chapter calculations and in equilibrium : pH questions and more topics .
• Logarithm function is also in used in Physics Calculations and Mathematics calculations .

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More About Logarithm function :

1) The $\sf\:f(x) =\log_{a}(x)\:and\:g(x)=a{}^{x}$

inverse of each other.So, the graph are mirror images of each other in the line mirror y = x.

2) Graphs of Logarithm function

$\sf\:f(x) =\log_{a}(x)$ when a > 1 in the attachment.