logarithms advanced

In the last article, we saw the basics of logarithms in terms of parts, bases and operations. While most of the time the questions require one to use these concepts and are straightforward yet a bit confusing, there are a few scorchers that can turn up from this area. These are the ones involving the number of digits in a particular expression, and the number of zeros following a decimal point in an expression. Let’s see how to tackle these types.

The basics remain the same. Just that you need to do a bit of reverse-engineering to arrive at the answer.

As was mentioned in the previous article the characteristic is nothing but the (number of digits of the number – 1). So, if a number has say 20 digits, the characteristic would be 19. Also, the fact that we use a 10-digit based system, you need to be able to express something in terms of 10x. This is the only bit of knowledge that is required. Beyond this, a bit of logic and application would be enough.

Although values of the required logs would be given (and in some cases, irrelevant logs might be given too to confuse you), it would be a good idea to know the logs of the first 10 natural numbers so that it becomes easier to solve:

Number Log (to the base 10) Number Log (to the base 10)
1 0 6 0.7782
2 0.3010 7 0.8450
3 0.4771 8 0.9031
4 0.6020 9 0.9542
5 0.6990 10 1


You will find that you can easily derive the values of the logs if we know the values of log 2, log 3, log 5 and log 7.

Let’s look at a few questions directly and understand the types:

Find the number of digits in 2800

Let a = 2800

log a = 800 log 2

log a = 800 * 0.3010

log a = 240.8

A common mistake done here is that the students get confused and mark 240 as the answer. However, if you have seen the nature of the characteristic, you will understand that if the value of log a is 240.xxxx, the number of digits in a would be 240+1 = 241.

Find the number of digits in 6500

Let a = 6500

log a = 500 log 6

log a = 500 (log 2 + log 3)

log a = 500 (0.3010 + 0.4771)

log a = 389.05

As the characteristic of log a is 389, the number of digits in a would be 390.

Another couple of question types, that follow the same logic could be as follows:

What power of 8 has 1000 digits?

In this, you know that 8a will have 1000 digits

Effectively, it means that 8a will lie between 10999 and 101000. The easiest way to deal with it is to take 8a = 10999

So, a log 8 = 999

a = 999/(log 8)

a = 1106.20

In this, we take the higher integer and so, it becomes 1107.

PS: A few of you might think what it would be like to not assume it to be 10999 but something else. However, you can also cross-check for 81106 and 81107 and you would get the number of digits to be 999 and 1000 respectively.

The final part of the topic involves questions requiring you to find the number of 0s after the decimal point in a certain expression. Let’s take an example and understand how to go about the questions.

Find the number of 0s that appear after the decimal point in the expansion 1/(2037)

This is a simple question. You can simply write it as

a = 1/(2037)

log a = -37 log 20

log a = -37 (1.3010)

log a = -48.1370

log a = bar(49.8630)

So, a has |-49+1| = 48 zeros.

So, the number of zeros post the decimal in a would be 48.

PS: You can visualize it from log (1/10) = -1.0000 and the fact that 1/10 = 0.1 and so, no zeros. So, as soon as you get something in the form of a bar, you simply add 1 to it and get the answer.

If log3 = 0.47712, log2 = 0.30103, determine how many zeros there are between the decimal point and the first significant digit in (2/3)100?

Again, you have to simply find the form of log (2/3)100

Approximately equal to 100 {log 2 – log 3} = 100 * (-0.17609) = -17.609 which is equal to bar(18.391) and so, number of zeros post the decimal point would be |-18+1| = 17.

Final thoughts

The question type can very well be expected in the test and probably more so in case of XAT or a one off question in IIFT. For a trained aspirant (and now, you!), it would classify as a sitter if anything related to the topic of logarithms turns up in CAT 2015. The questions are single concept based and in most cases will give you an answer in 4-5 steps and are the differentiators between someone who scores say a 95%ile and someone who goes that bit extra and scores a 99+ score.

All the best!

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