Change of Base Formula for Logarithms

Logarithm is a means of expressing a number using exponents. Example log101000 is equal to 3 as 1000 is a cube of ten and can be written as log_10 10^3. Hence the value is 3.
The common base for logarithms is base ten and the other base is the natural logarithm base –e. At times while calculating logarithms we come across base other than 10 and the base e, in such cases the base change can be done using a special formula.

Logarithm change of base formula can be given as, log x to base a = log x to base b/log a to base b.  To understand how to arrive to this base change formula let us go through the following steps:
Consider y=log_a x, we get x = a^y
Taking log_b on both sides would result in log_b x = log_b a^y
Applying the power rule to the above equation gives, log_b x = y log_b a
Now dividing on both sides with log_b a gives, log_b x/ log_b a= y log_b a/ log_b a
So, we get, y = log_b x/ log_b a

Let us now consider a simple example, the value of log 27. This can be written as log_10 7/log_10 2

The value can be calculated as log 7=0.845 and log2= 0.3010. When these values are divided the final answer would be 2.80730…; thus using loga x= log_bx/log_b a, the change of base formula logarithms value of the given logarithmic expression can be found easily. Using Log base change formula it becomes easy to evaluate logarithms with different base. Here the logarithm is written as a fraction with the logarithm of the number as the numerator and the logarithm of the base as the denominator, such as log_a x = log x/log a.

Then each of the logarithms is evaluated using the log table or a scientific calculator, the final value is got by dividing these values. The evaluation of other logarithms with base different from natural logarithm base or the common logarithm base can be done using the base change formula, log_a x = log_b x/log_b a. Let us now evaluate the logarithm log_5 9. This problem can be solved by either using natural logarithm or the common logarithm. Using the natural logarithm that is base-e it would be, log_5 9 = ln9/ln5 = 2.1972/1.6094 which would be approximately equal to 1.3652… Now using the common logarithm that is the base ten it would be, log_5 9 = log 9/log5= 0.9542/0.6989 = 1.3652… Using either of the logarithms we arrive at the same result.

Simple interest

Definition:
Consider a house that one would have rented. The tenant has to pay some amount of money to the owner of the house as rent for using the property. Similarly if a person borrows money from another person, he has to pay some amount of money as rent for using the borrowed money. This charge paid for use of funds is called interest. Therefore the amount charged on a fixed amount of principal, that is lent by a lender for a specific period of time is called simple interest. In simple interest the principle amount over the period of loan remains constant and is not reduced or increased.
Formula for simple interest:
Some important terms related to simple interest:
(1) Principal (P): The money borrowed or lent.
(2) Interest (I): The additional amount paid to the lender, for the use of the money borrowed.
(3) Rate( R ): Interest for one year per 100 units of currency.
(4) Time (T): The time period for which the money is borrowed.
(5) Simple interest or (S.I.): When the interest is paid to the lender regularly every year or every half year, we call the interest simple interest.
(6) Amount (A): Principal + Interest = amount at the end of the term of T years.

Formula used for calculating simple interest is like this:
S.I. = P x R x T
100
A = P + S.I.

When we calculate simple interest, the following points need to be noted:
(1) Rate of 4% per annum means $ 4 for every $ 100 per year. Similarly a rate of 1.5% per month means $ 1.5 for every $ 100 per month = $ 1.5 * 12 = $ 18 for every $ 100 per  year = 18% per annum.
(2) When time is given in days, we convert it to years by dividing by 365. When time is given in  months, we convert it to years by dividing by 12. When dates are given, the day on which the sum is borrowed is not included but the day on which the money is returned is included, while counting the number of days.

Decimal to Hexadecimal

We are now going to look at Decimal to Hexadecimal converter. So let us understand what exactly a hexadecimal number and what its digits mean. So we are going to look at three digits of hexadecimal number the first unit represents units, which is 16 to the power of zero that is one. That represents units. The second digit represents tenths, which is 16 to the power of one. And the third digit represents hundredths, which is 16 to the power of two. That is nothing but 256. So important thing to do when one is working on how to convert a decimal to hexadecimal, is the start of working out how many hexadecimal number is going to have?

Let us understand it with an example, convert decimal to hexadecimal. Say number 74, here we need to decide, what we are going to and how many digits this hexadecimal number is going to have. Now because 256 is less than 74, there is any going to be two digits. So we now going to see, that in 16 to the power of one column, here we divide 74 by 16 and the result is 4. This means 4 times 16 is 64 and we have the remainder as 10. Now 10 is a single digit in a hexadecimal, simply represents a A , that tells us 74  = 4 A.

 Let us understand with a complicated example. This time it represents 680 as a hexadecimal number. Here that we see 680 is greater than 256 so we are going to have three digits in a hexadecimal number. What we going to do first is divide 680 by 256. And the result of that is 2. Two times 256 is 512, so our remainder is 168. Next we go back as what we did in our first example, we are going to divide 168 by 16. The result of this is 10. 10 times 16 equals to 160, as we are left with the remainder 8. Now we have three digits in hexadecimal number, thus we notice that we have 10, which is represented by ( A ) . This tells us that 680 when written as a hexadecimal number as 2A8. That is, 680 = 2A8. This is how we do a decimal to hexadecimal conversion. The method to do this is to keep on dividing the decimal number by 16 till it gets the most significant remainder.

Rules of Narration for Different Types of Sentences

Narration is one of the most important concepts in English grammar. While changing narration, it is very important to follow certain rules. These rules at times differ according to the types of sentences. Let’s have a look at the rules of narration for different types of sentences in this post.
Rules of Narration for Assertive Sentences:

Rule 1: If there is no object after reporting verb, then it should not be changed. For example:
• Direct Speech: He said, “I bought a play gun from Nerf India collection for my nephew.”
Indirect Speech: He said that he bought a play gun from Nerf India collection for his nephew.
Rule 2: If there is any object after the reporting verb, then say is changed to tell, ‘says’ to ‘tells’ and ‘said’ to ‘told’. For example:
• Direct Speech: She said to me, “Pre Nan Nestle Baby is healthy and nutritious for babies.”
Indirect Speech: She told me that Pre Nan Nestle is healthy and nutritious for babies.
Rule 3: ‘said’ can be replaced by replied, stated, and added and more as per the context of the assertive sentence. For example:
• Direct Speech: She said to him, “I am going to school today.”
Indirect Speech: She replied to him that she is going to school that day.
Rules of Narration for Interrogative Sentences:
Rule 1: In interrogative sentences, ‘said’ is changed to ‘asked’ while changing from direct to indirect speech. At times, ‘said’ is also changed to ‘enquired’ or related terms as per the context.
Rule 2: If the question is formed with is/are/am/was/were etc. then it is replaced by ‘if’ or ‘whether’.
Rule 3: While changing from direct to indirect speech, the question mark is removed as the reported speech is an indirect statement and not a direct question.
For example:
• Direct Speech: She said to him, “Have you bought anything from Philips Avent India brand?”
Indirect Speech: She asked to him whether he has bought anything from Philips Avent India brand.
These are some of the rules of narration that is defined as per different types of sentences.

Interjection and its types

Interjection is one of the eight parts of speech in English grammar along with noun, pronoun, verb, adverb, adjective, conjunction and preposition. Interjection is a part of speech that conveys emotions or expresses a meaning or feelings. Interjection does not add meaning to a sentence but express the feeling of the speaker. An interjection is sometimes followed by the exclamation sign i.e. “!”. For example: The kid exclaimed, “Hurrah! I got Barbie coloring pack of pens”. Here, the interjection “hurrah!” is expressing the excitement of the kid after getting Barbie coloring pack of pens. There are five different types of interjections classified based on expressions such as greetings, joy, approval, surprise and grief. Lets’ have a closer look at each of the types of interjections.

Interjections to express Greetings:
Interjections to express greetings are type of interjections that are used to wish someone. Popularly used interjections of expressing greetings are: hello, hi and so on. For example: Hello, how have you been? Here, the speaker is wishing hello and how the other person has been by using “hello”.

Interjections to express Joy:
As the term suggests, these types of interjections convey the feeling of joy and excitement of the speaker. Commonly used interjections are: hurrah, yippee, hey and more. For example: Hey! Look Farlin India collection has real good stuffs for kids. Here, the speaker is expressing his excitement at Farlin India brand’s grand collection.

Interjections to express Approval:
Interjections to express approval are used to express or congratulate on someone’s effort. Commonly used interjections of expressing approval are: bravo, wow etc. For example: Bravo! You won the match. Here, the speaker is congratulating some for winning the match.

Interjections to express Surprise:
Interjections to express surprise are used to express the emotion of surprise by the speaker. Commonly used interjections of expressing surprise are: oh, eh and so on. For example: Oh! Nuby baby brand is giving away annual sale this month. Here, the speaker is expressing his surprise at Nuby baby brand’s annual discount.

Interjections to express Grief:
Interjections to express grief as the term describes is used to express sorrow or grief by the speaker. Commonly used interjections expressing grief are: alas, ouch and more. For example: Alas! The man is dead. Here, the speaker is using “Alas” to express his grief on the man’s death.