Important Concepts and Formulas - Logarithm

ssc preparation, important competetion questions, current affairs of india, latest question asked in competetion papers,

Important Concepts and Formulas - Logarithm1. Basics

$\text{if}y={\log }_{b}x\text{, then}{b}^y=x$

where ${\log }_{b}x=y$ = log to the base $b$ of $x$
Please note that $b$(base) is a positive real number, other than 1.

$\text{if}x=b^y\text{, then}{\log }_{b}x=y$

where ${\log }_{b}x=y$ = log to the base $b$ of $x$
Please note that $b$(base) is a positive real number, other than 1.

Example

$16=2^4$ (in this expression, 4 is the power or the exponent or the index and 2 is the base)
Hence we can say that ${\log }_{2}16=4$ (i.e., log to the base 2 of 16 = 4)
In other words, both $16=2^4$ and ${\log }_{2}16=4$ are equivalent expressions.

2. Common and Natural Logarithm

If base = 10, then we can write $\log x$ instead of ${\log }_{10}x$
$\log x$ is called as the common logarithm of $x$

If base $=e$, then we can write $\ln x$ instead of ${\log }_{e}x$
$\ln x$ is called as the natural logarithm of $x$

Please note that $e$ is a mathematical constant which is the base of the natural logarithm. It is known as Euler's number. It is also called as Napier's constant.

$e=1+\frac{1}{1}+\frac{1}{1.2}+\frac{1}{\mathrm{1.2.3}}$ $+\frac{1}{\mathrm{1.2.3.4}}+\cdots \approx 2.71828$

$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots$

3. Logarithms - Important Properties${\log }_{b}1=0$        (∵ $b^0=1$)

${\log }_{b}b=1$        (∵ $b^1=b$)

$y=\ln x\Rightarrow e^y=x$

$x=e^y\Rightarrow \ln x=y$

$x=\ln e^x=e^{\ln x}$

$b^{{\log }_{b}x}=x$

${\log }_b{b}^y=y$

4. Laws of Logarithms1. ${\log }_b\text{MN}={\log }_b\text{M}+{\log }_b\text{N}$ (where b, M, N are positive real numbers and b ≠ 1)

2. ${\log }_b\frac{\text{M}}{\text{N}}={\log }_b\text{M}-{\log }_b\text{N}$(where b, M, N are positive real numbers and b ≠ 1)

3. ${\log }_b{M}^c=c{\log }_b\text{M}$ (where b and M are positive real numbers , b ≠ 1, c is any real number)

4. ${\log }_{b}M=\frac{\log \text{M}}{\log b}=\frac{\ln \text{M}}{\ln b}=\frac{{\log }_k\text{M}}{{\log }_{k}b}$ (where b, k and M are positive real numbers, b ≠ 1, k ≠ 1)

5. ${\log }_{b}a=\frac{1}{{\log }_{a}b}$ (where a and b are positive real numbers, a ≠ 1, b ≠ 1)

6. If ${\log }_{b}M={\log }_b\text{N}$, then M = N (where b, M and N are positive real numbers and b ≠ 1).

5. Mantissa and CharacteristicThe logarithm of a number has two parts, known as characteristic and mantissa.

1. Characteristic
The internal part of the logarithm of a number is called its characteristic.

Case I: When the number is greater than 1.
In this case, the characteristic is one less than the number of digits in the left of the decimal point in the given number.

Case II: When the number is less than 1.
In this case, the characteristic is one more than the number of zeros between the decimal point and the first significant digit of the number and it is negative. Instead of -1, -2 etc. we write $\overline{1}$(one bar), $\overline{2}$ (two bar), etc.
Examples

Number	Characteristic
612.25	2
16.291	1
2.1854	0
0.9413	$\overline{1}$
0.03754	$\overline{2}$
0.00235	$\overline{3}$

2. Mantissa
The decimal part of the logarithm of a number is known is its mantissa. We normally find mantissa from the log table.