Applied Mathematics

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

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:

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

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

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

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

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

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 ₄₃57

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

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

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

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Introduction to logarithms

Applied Mathematics

Introduction to logarithms

Learning Objectives of Teaching & Learning Logarithms

Learning Outcomes of Teaching & Learning Logarithms

Fundamental Properties of Logarithms

Characteristic and Mantissa

Rule to find the Characteristic

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