# Higher Algebra Hall & Knight Logarithms (Chapter 16) Solutions

Hall and Knight Higher Algebra Solutions for Chapter 16 ‘Logarithms’ are useful for aspirants preparing for engineering entrance exams like JEE. These solutions include details about logarithms, finding the log of a product, finding the log of a fraction, finding the log of any number raised to any power, characteristics, and mantissa of the logarithms. They consist of comprehensive explanations along with solved exercise questions of the chapter to keep all the concepts clear for you along with some quick formulas for JEE.

Higher Algebra By Hall and Knight Logarithms contain two exercises with around 56 questions. In these advanced level questions, you need to find the values of log-like loge128 or expressing the logarithms in terms of log a, log b and log c, finding the characteristics of logarithms and mantissa, calculating the equation to six decimal places and much more. The questions have been designed by Hall and Knight in a manner that they make the preparation for engineering entrance exams easy as all the types of questions that can be asked in JEE Mains and JEE Advanced are covered in the exercises.

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## Important Topics for Higher Algebra by Hall & Knight Chapter 16: Logarithms

The various topics covered under the chapter are discussed below:

** Logarithm**

The logarithm is known as the inverse function to exponentiation. This means that the logarithm of a number x is mainly the exponent to which another number, the base b, that is fixed must be raised in order to produce that number x.

In simple words, the logarithm is all about how many times we multiply a number to get another number.

Basically, the number we multiply is the base. For example, if we write 4 x 4 x 4= 64, then the logarithm of 64 with base 4 is 3 or log base 4 of 27 is 3.

**How to find the Log of a Product**

The product rule for logarithms is ln(xy) = ln(x) + ln(y). You are required to calculate the log of a product using the given formula.

**How to find the Log of a Fraction**

The log of a fraction can be calculated by evaluating the logarithm of the numerator and denominator. For example: ln(x/y) = ln(x) – ln(y). Applying these to a fraction we get ln(x/y) = ln(xy^{–}^{1}) = ln(x) + ln(y^{–}^{1}) = ln(x) – ln(y).

**How to find the Log of Any Number Raised to Any Power**

If any number is raised to the power of a logarithm, that number is its base which is equivalent to the number that you used in the logarithm.

The power rule of the logarithm can be used for lengthening and precipitating logarithms.

**Characteristics and Mantissa**

- Inspection and the mantissa by the logarithm table determine the characteristic of a logarithm of a number.
- The characteristic of a logarithm is an integral part of a common logarithm and mantissa is the non-negative part with decimal.
- You have to find the characteristics of the logarithm in various situations like when the logarithm of a number is greater than 1, a number greater than 1 is positive and is 1 less than the number of two digits, the logarithm of a number lying between 0 and 1, and the logarithm of a number that is positive less than 1 is negative and is numerically greater by 1 than the number of zeros between the decimal and the first important figure of the number.
- To find the mantissa we have to go through the logarithmic table.
- There are four and five-figure tables. A four-figure table determines the value of mantissa correct to 4 decimal places.

### Exercise Discussion of Higher Algebra by Hall and Knight Chapter 16: Logarithms

- Hall & Knight Algebra Book Chapter 16 ‘Logarithms’ comprises 2 exercises with overall 56 questions that are unsolved.
- Both the exercises contain authentics questions which help you to practice the major and complex concepts of the chapter logarithm.
- The first exercise has 23 questions. These questions are based on vital concepts like calculating log of a fraction or log a product. The questions that you will solve are like find the value of log
_{8}128 and log_{6}1/216, and some basic questions like finding the logarithms of 0625 to base 2, and 1000 to base 01. - In the second exercise, there are 32 questions and you will have to find the solutions for the questions such as: find the seventh root of 00324 when the log is given, calculate to six decimal places the value of the given equation, find the number of digits in 3
^{12}and 2^{8}and many other questions.

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