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### Introduction to logarithms

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__Introduction to logarithms__**Simple explanations and the implementations of logarithmic formulas in solving complicated mathematical problems. Indices, logarithmic formulas and logarithmic tables are explained here**

**Logarithmic Tables**

**********************

__Indices__

**x ^{m}
= x.x.x.x…………….m terms**

**Here x is
called the base and m is called the index or exponent of x.**

**Indices is
the plural of index.**

**\[If\;\; \;
\; x^{2}=a\; \; \; \Rightarrow\; \; \; x=\sqrt{a}\]**

**\[If\;\; \;
\; x^{3}=a\; \; \; \Rightarrow\; \; \; x= \sqrt[3]{a}\]**

**\[If\;\; \;
\; x^{4}=a\; \; \; \Rightarrow\; \; \; x= \sqrt[4]{a}\]**

**\[If\;\; \;
\; x^{5}=a\; \; \; \Rightarrow\; \; \; x= \sqrt[5]{a}\]**

**\[If\;\; \;
\; x^{100}=a\; \; \; \Rightarrow\; \; \; x= \sqrt[100]{a}\]**

**\[If\;\; \;
\; x^{n}=a\; \; \; \Rightarrow\; \; \; x= \sqrt[n]{a}\]**

__Laws of
Indices__

**\[(i)\; \;
x^{m}\times x^{n}=x^{m+n}\; \; \; (ii)\; \; \; \; \frac{x^{m}}{x^{n}}=x^{m-n}\]**

**\[(iii)\;
\; (x^{m})^{n}=x^{mn}\; \; \; (iv)\; \; \; \; x^{m}=\frac{1}{x^{-m}}\; \; or\;
\; x^{-m}=\frac{1}{x^{m}}\]**

**\[(v)\; \;
\; \; (xy)^{m}=x^{m}y^{m}\; \;\: \; (vi)\; \; \; \left ( \frac{x}{y} \right
)^{m}=\frac{x^{m}}{x^{n}}\]**

**\[(vii)\;
\; \; \; x^{a}=y^{a}\; \;\Rightarrow \: \;x=y,\; \;Provided\; \; a\neq 0\]**

**\[(vii)\;
\; \; \; x^{m}=x^{n}\; \;\Rightarrow \: \;m=n,\; \;Provided\; \; x\neq \: 0,1\]**

__Logarithms__

**Logarithm of a number to a given base is the index or power to which the base must be raised so as to be equal to the given number.**

**If a ^{x} = N (Where a is a positive number ≠1), then x is called the logarithm of N to the base a and is written as x = Log_{a} N.**

**\[For\; Example:\; \; 5^{3}=125, \Rightarrow 3=Log_{5}125\]**

**\[ 4^{5}=1024, \Rightarrow 5=Log_{4}1024\]**

**\[Log1=0,\; \; log10=1, \; \; log100=2,\; \; log1000=3\; \; ....so \; \; on.\]**

**\[Log_{a}1=\frac{Log1}{Loga}=\frac{0}{Loga}=0\]**

**\[Log_{a}a=\frac{Loga}{Loga}=1\]**

**Example:**** \[If\; \; x = 2^{-\frac{1}{3}log_{2}64},\; \; then\; find\; \; x\]**

**Solution:**** \[x = 2^{-\frac{1}{3}log_{2}64}\; \; =2^{log_{2}\left ( 64 \right )^{-1/3}}\]**

**\[=2^{log_{2}\left ( 4^{3} \right )^{-1/3}}=2^{log_{2}(4)^{-1}}\]**

**\[=2^{log_{2}(2^{2})^{-1}}=2^{log_{2}(2)^{-2}}=2^{-2log_{2}2}\]**

**\[=2^{-2\times 1}=2^{-2}=\frac{1}{2^{2}}=\frac{1}{4}\]**

**Fundamental Properties of Logarithms**

**Product Formula:**** \[Log_{a}(lm)=Log_{a}l+Log_{a}m\]**

**\[Log_{a}(lmn)=Log_{a}\: l+Log_{a}\: m+Log_{a}\: n\]**

**Quotient Formula:**** \[Log_{a}(\frac{m}{n})=Log_{a}\: m-Log_{a}\: n\]**

**Power formula:**** \[Log_{a}m^{n}=n\: Log_{a}\: m\]**

**Example:**** \[Evaluate:\: \: Log_{5}\; \left ( \frac{\sqrt[4]{25}}{625} \right )\]**

**Solution: \[Log_{5}\; \left ( \frac{\sqrt[4]{25}}{625} \right )=Log_{5}\sqrt[4]{25}-Log_{5}\; 625\]**

**\[=Log_{5}\; (25)^{1/4}-Log_{5}\; 5^{4}=\frac{1}{4}Log_{5}\; (25)-4Log_{5}5\]**

**\[=\frac{1}{4}Log_{5}\; (5)^{2}-4\times1 =\frac{1}{4}\times 2\; Log_{5}\; (5)-4\]**

**\[=\frac{1}{2}\times 1-4=\frac{-7}{2}\]**

**Natural Logarithms**

**Logarithms to the base e (= 2.7183) are called natural logarithms. They are used in all theoretical calculations. For Example : Log**

_{e }(sin x) This logarithm is called the natural logarithm.

Common Logarithms

Logarithms to the base 10 are called common logarithms. They are used in arithmetic calculations. For Example : Log

Common Logarithms

Logarithms to the base 10 are called common logarithms. They are used in arithmetic calculations. For Example : Log

_{10 }(34.56) This logarithm is called the common logarithm.

Characteristic and Mantissa

Characteristic and Mantissa

The integral part of the logarithm of a number, after expressing the decimal part as positive (if not already so), is called the characteristic.

Mantissa

Positive decimal part is called Mantissa. Note: Mantissa is always positive.

The integral part of the logarithm of a number, after expressing the decimal part as positive (if not already so), is called the characteristic.

Mantissa

Positive decimal part is called Mantissa. Note: Mantissa is always positive.

**Rule to find the Characteristic**

Rule 1

Rule 1

**Characteristic of the logarithm of a number > 1 is positive****Characteristic of the logarithm of a number is one less than the number of digits in the integral part of the number.**

**For Example : 34.586**

**Integral Part Before Decimal = 34****No. Digits in the integral part = 2****Characteristic = 2 - 1 = 1**

**Rule II**

**Characteristic of the logarithm of a number < 1 is negative.****It is numerically one more than the number of zeroes immediately after the decimal point.**

**For Example : 0.0586 **

**Here given number is < 1 so characteristic should be negative**

**Number of zeroes after decimal = 1**

**Characteristic = - (1 + 1) = - 2**

**Negative character always written in bar form as given below \[-2 \: \: is\: written\: in\: the\: form\: of\: bar\: as\: \: \overline{2}\]**

**Characteristic and Mantissa for a negative number**

**For Example:**

**Let the number is - 2.305**

**- 2.305 = - (2 + 0. 305) = - 2 - 0.305**

**To make decimal part positive we have to subtract and add 1 to this number**

**- 2 - 1 + 1 - 0.305 = - 3 + 0.695 \[=\overline{3}.695\]**

**Find the number of digits in 3**

^{62}**Let x
= 3 ^{62}**

**Taking log on both side**

**Log x = Log 3 ^{62}**

**Log x = 62 Log 3 = 62 X 0.4771 = 29.5802**

**Characteristic
= 29**

**Number of digits in 3 ^{62} = 29 + 1 = 30**

**Find the position of the first significant figure in 3**

^{-65}**Let x = 3 ^{-65} **

**Log x = Log 3 ^{-65} = - 65 X Log 3
= -65 X 0.4771 = -31.0115**

**Log x = -(31 + 0.0115) = -31 – 0.0115**

**In order to make Mantissa positive , add and subtract 1**

**Log x =
- 31 – 1 + 1 -0.0115**

**
= - 32 + 0.9985**

**Characteristic Value = - 32**

**So there are 32 – 1= 31 zeroes immediate after
decimal.**

**So position of the first significant figure
is 31 + 1 = 32**

**Find the 5th root of 0.003**

**\[Let\: \: x=\left ( 0.003 \right )^{1/5}\]**

**\[Logx=Log\left ( 0.003 \right )^{1/5}\]**

**\[Log x=\frac{1}{5}\; Log \left ( 0.003 \right )\]**

**\[Log x=\frac{1}{5}\times \overline{3}.4771\: =\: \frac{1}{5}[-3+0.4771]\]**

**\[Log x=\: \frac{1}{5}[-3-2+2+0.4771]=\: \frac{1}{5}[-5+2.4771]\]**

**\[Log x=\: \frac{1}{5}\times -5+\frac{1}{5}\times 2.4771 = -1+0.4954 = \overline {1} .4954\]**

**Taking ****antilog**** on both sides we get**

**\[x=Antilog(\overline{1}.4954)=0.3129\]**

**Find the value of**

**Log**

_{43 }57**\[Let \; \; x = Log_{43}57 = \frac{Log57}{Log43}=\frac{1.7559}{1.6335}\]**

**\[Log x = Log\left (\frac{1.7559}{1.6335} \right )=Log1.7559-Log1.6335\]**

**\[Log x = 0.2443-0.2130=0.0312\]**

**Taking antilog on both sides we get**

**\[x = Antilog\: (0.0312)=1.074\]**

**Evaluate the following**

**\[Evaluate\: :\: \sqrt[3]{\frac{(45.4)^{2}}{(3.2)^{2}\times (6.5)^{3}}}\]**

**Solution : \[Let\; \; x=\: \sqrt[3]{\frac{(45.4)^{2}}{(3.2)^{2}\times (6.5)^{3}}}\]**

**Taking log on both side we get**

**\[Log\: x=Log \sqrt[3]{\frac{(45.4)^{2}}{(3.2)^{2}\times (6.5)^{3}}}\]**

**\[Log\: x=Log \left (\frac{(45.4)^{2}}{(3.2)^{2}\times (6.5)^{3}} \right )^{1/3}\]**

**\[Log\: x=\frac{1}{3}Log \left (\frac{(45.4)^{2}}{(3.2)^{2}\times (6.5)^{3}} \right )\]**

**\[Log\: x=\frac{1}{3}\left [Log(45.4)^{2}-log(3.2)^{2}-log(6.5)^{3} \right ]\]**

**\[Log\: x=\frac{1}{3}\left [2Log(45.4)-2log(3.2)-3log(6.5) \right ]\]**

**\[Log\: x=\frac{1}{3}\left [2\times 1.6571-2\times0.5051-3\times 0.8129 \right ]\]**

**\[Log\: x=\frac{1}{3}\left [3.3142-1.0102-2.4387 \right ]\]**

**\[Log\: x=\frac{1}{3}\left [3.3142-3.4489 \right ]\]**

**\[Log\: x=\frac{1}{3}\times \left (-0.1347 \right )\]**

**\[Log\: x=\frac{1}{3}\times \left (-3+3-0.1347 \right )=\frac{1}{3}\times \left (-3+2.8653 \right )\]**

**\[Log\: x=-1+0.9551=\overline{1}.9551\]**

**Taking Antilog on both side we get**

**\[x=Antilog\left (\overline{1}.9551 \right )\]**

**\[x=Antilog\left (\overline{1}.9551 \right ) = 0.9018\; Ans\]**

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