Mathematics Assignments | PDF | 8 to 12

PDF Files of Mathematics Assignments From VIII Standard to XII Standard PDF of mathematics Assignments for the students from VIII standard to XII standard.These assignments are strictly according to the CBSE and DAV Board Final question Papers

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
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Learning Objectives of Teaching & Learning Logarithms

Students should study the logarithms to achieve the following objectives:-

Understanding the concept of logarithms:

The primary objective is to help students understand what logarithms are and how they relate to exponentiation.

Solving logarithmic equations:

Students should learn how to solve logarithmic equations, including properties of logarithms such as the product, quotient, and power rules.

Applying logarithms to real-world problems:

Students should be able to apply logarithms to solve real-world problems, such as exponential growth and decay, compound interest, and logarithmic scales.

Learning Outcomes of Teaching & Learning Logarithms

After study of this topic students should possess the following skills

Computational skills:

Students should be able to perform calculations involving logarithms, such as evaluating logarithmic expressions, simplifying logarithmic equations, and solving logarithmic equations for unknowns.

Problem-solving skills:

Students should be able to apply logarithms to solve a variety of mathematical and real-world problems, including those involving exponential growth, decay, and rates of change.

Critical thinking and reasoning:

Students should develop critical thinking skills by analyzing and interpreting logarithmic expressions and equations in different contexts, and making connections between logarithms and exponentiation.

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:

$5^{3}=125, \Rightarrow 3=Log_{5}125$

$4^{5}=1024, \Rightarrow 5=Log_{4}1024$

$Log1=0,\; \; log10=1, \; \; log100=2,\; \; log1000=3\; \; ....so \; \; on.$

$Log_{a}1=\frac{Log1}{Loga}=\frac{0}{Loga}=0$

$Log_{a}a=\frac{Loga}{Loga}=1$

Question:  $If\; \; x = 2^{-\frac{1}{3}log_{2}64},\; \; then\; find\; \; x$

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 Rule:

$Log_{a}(lm)=Log_{a}l+Log_{a}m$

$Log_{a}(lmn)=Log_{a}\: l+Log_{a}\: m+Log_{a}\: n$

Quotient Rule:

$Log_{a}(\frac{m}{n})=Log_{a}\: m-Log_{a}\: n$

Power Rule:

$Log_{a}m^{n}=n\: Log_{a}\: m$

Question:

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

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 10 (34.56)  This logarithm is called the common logarithm.

Characteristic and Mantissa

Characteristic
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
• 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
$=\overline{3}.695$
Question:  Find the number of digits in  362

Solution

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

Question:  Find the position of the first significant figure in  3-65

Solution

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

Question:  Find the 5th root of 0.003
Solution:

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

$Logx=Log\left ( 0.003 \right )^{1/5}$

$Log x=\frac{1}{5}\; Log \left ( 0.003 \right )$

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

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

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

Taking antilog on both sides we get

$x=Antilog(\overline{1}.4954)=0.3129$

Question: Find the value of  Log 43 57

Solution:

$Let \; \; x = Log_{43}57 = \frac{Log57}{Log43}=\frac{1.7559}{1.6335}$

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

$Log x = 0.2443-0.2130=0.0312$

Taking antilog on both sides we get

x = Antilog(0.0312) = 1.074

Evaluate the following

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

Solution :

$Let\;\;x=\:\sqrt[3]{\frac{(45.4)^{2}}{(3.2)^{2}\times(6.5)^{3}}}$
Taking log on both side we get

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

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

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

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

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

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

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

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

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

$Log\:x=-1+0.9551=\overline{1}.9551$
Taking Antilog on both side we get

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