Logarithm

The inverse function of the exponentiation is the logarithm. For any given number, ‘a’ is termed as the exponent of the other number consisting of base ‘b’ that is not a variable, should be raised to get the number a is called the logarithm of that number. The concept of the logarithm can be referred to as the sum of the number of frequencies of the same factor in the form of repeated multiplication. For example, 1000 can be expressed as 10 multiplied 3 times.

1000 = 10 * 10 * 10 = 10³

The logarithmic expression of 1000 (to base 10) can be written as log₁₀ (1000) = 3.

The concept of logarithms was established by a Scottish mathematician, physicist, and astronomer named John Napier in 1614. It was extensively used by engineers, surveyors, scientists in computations that involved high accuracy.

Using logarithms, multi-digit multiplication steps can be deduced into simpler ones by implementing logarithm tables. The most frequently used logarithm is to base 10. The integral part of the logarithm is the characteristic and mantissa is the decimal part. It is represented as follows.

log N = Characteristic + Mantissa (positive)

Properties of logarithms are given below.

1. log_a (mn) = log_a m + log_a n
2. log_a (m / n) = log_a m – log_a n
3. log_a m^q = q log_a m
4. log_b m = log_a m / log_a b
5. log_b a . log_a b = 1 ⇒ log_b a = 1 / log_a b
6. log_b a . log_c b . log_a c = 1
7. log_y x . log_z y . log_a z = log_a x
8. e^{ln a^x} = a^x

Few applications of logarithms are listed below.

• The theory of scales of a logarithm is used in lowering down the wide range of tiny scopes.
• The acidity of an aqueous solution called pH can be measured using them.
• To quantify the difficulty of algorithms and objects of geometry.
• The ratio of the frequency of intervals of music can be explained through logarithms.
• Used in forensic accounting.

The students facing JEE examination will find the important questions for JEE Main helpful as it enables them to predict the type of questions that can appear in the final examination. It also assists them in understanding the concepts in a better way. The previous year subject and chapterwise questions and solutions as the most reliable study material which covers all the essential topics and chapters. The solved questions consist of detailed explanation along with the hints which benefit the students.

Previous Year JEE Questions On Logarithms

Question 1: If 3^x = 4^x-1, then x =

Solution:

3^x = 4^x-1

Taking log on both sides with base a,

x log_a 3 = (x – 1) log_a 4

x = [log_a 4] / [log_a 4 – log_a 3]

If a = 3: x = [2 log₃ 2] / [2 log₃ 2 – 1]

If a = 2: x = [2] / [2 – log₂ 3]

If a = 4: x = [1] / [1 – log₄ 3]

Question 2: Find the value of x satisfying log₁₀ (2^x + x – 41) = x (1 – log₁₀5).

Solution:

The given expression is

log₁₀ (2^x + x – 41) = x (1 – log₁₀5)

log₁₀ (2^x + x – 41)  = x log₁₀ 2 = log₁₀ (2^x)

2^x + x – 41 = 2^x

x = 41