Simple explanations and the implementations of logarithmic formulas in solving complicated mathematical problems. Indices, logarithmic formulas and logarithmic tables are explained here
Indices
xm
= 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 = Loga 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
Find the number of digits in 362
Let x
= 362
Taking log on both side
Log x = Log 362
Log x = 62 Log 3 = 62 X 0.4771 = 29.5802
Characteristic
= 29
Number of digits in 362 = 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 : \[Let\; \; x=\: \sqrt[3]{\frac{(45.4)^{2}}{(3.2)^{2}\times (6.5)^{3}}}\]
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