Logarithms | Quantitative Aptitude for CA Foundation PDF Download

Logarithm 

The logarithm of a number to a given base is the index or the power to which the base must be raised to obtain that number.

If am  = b (where b > 0, a ≠ 1, a > 0), then the exponent m is said to be logarithm of b to the base a.

So, if am = b       (Exponent form) 

Then, m = loga b   (Logarithm form)

Example 1. Express in logarithmic form: (i) 23  (ii) xm = y

Solution : 23 = 8 ⇒ log28 = 3

xm = y ⇒ logxy = m

Note: 

  • Logarithm of any quantity to the same base is 1. 
  • Logarithm of 1 to any base is zero

Types of Logarithm

1. Logarithms to the base “e” (exponential constant) are called natural logarithms.

2. Logarithms to the base “10” are called common logarithms

Note: If the base is not given, it can be assumed to be 10

Laws of Logarithm

1. loga(mn) = loga m + loga n

Logarithm of a product of two numbers can be written as sum of logarithms of the individual numbers.

Example 1.  If log10 2 = 0.3010, log10 3 = 0.4771, find log10 6.

Solution : log10 6 =  log10 (2x3)

log10 2=  log10 3

= 0.3010 + 0.4771 = 0.7781

Example 2. Simplify log 2 + log 3 + log 4

Solution : log 2 + log 3 + log 4 = log (2 × 3 × 4) = log 24

2. Logarithms | Quantitative Aptitude for CA Foundation

Logarithm of a division of two numbers can be written as difference of logarithms of the individual numbers.

Example:  Logarithms | Quantitative Aptitude for CA Foundation

3. (i) logamn = n logam

(ii) Logarithms | Quantitative Aptitude for CA Foundation

Logarithm of the nth  power of a number is n times the logarithm of that number.

Logarithm of a number m to the base a raised to power n is equal to 1/n times the logarithm of m to the base a.

Example 1 :  If log10 2 = 0.3010, find log10 8.

Solution : log108 = log102 = 3 log10 2 = 3 × 0.3010 = 0.9030

Example 2.  Find the value of 1/2log10 = 100

Solution : 

Logarithms | Quantitative Aptitude for CA Foundation
Logarithms | Quantitative Aptitude for CA Foundation

4. Logarithms | Quantitative Aptitude for CA Foundation

Example : Logarithms | Quantitative Aptitude for CA Foundation

Solution : Logarithms | Quantitative Aptitude for CA Foundation

Logarithms | Quantitative Aptitude for CA Foundation

Logarithms | Quantitative Aptitude for CA Foundation  

= log10(2) 

 

5.  Logarithms | Quantitative Aptitude for CA Foundation

We can write any logarithm as a division of two logarithms (of the number and the base) taken at any common base. This theorem is very useful in solving problems having logarithms with many different bases.If we put m = a in the above result, we get another important result which is logn a × loga n = 1

Example : Find the value of log3 7 x log49 243

Solution :  To simplify, we will make use of change of base theorem and convert each term into a logarithm with common base of 10.

Logarithms | Quantitative Aptitude for CA Foundation

Logarithms | Quantitative Aptitude for CA Foundation

Logarithms | Quantitative Aptitude for CA Foundation

6. If loga m = x, then

a. log(1/a)  m = -x 

b.  Logarithms | Quantitative Aptitude for CA Foundation

c. Logarithms | Quantitative Aptitude for CA Foundation

The document Logarithms | Quantitative Aptitude for CA Foundation is a part of the CA Foundation Course Quantitative Aptitude for CA Foundation.
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FAQs on Logarithms - Quantitative Aptitude for CA Foundation

1. What is a logarithm and how is it used in mathematics?
A logarithm is a mathematical function that represents the exponent to which a given base must be raised to obtain a specific number. In other words, it is the inverse operation of exponentiation. Logarithms are used in various fields of mathematics, such as solving exponential equations, measuring the intensity of earthquakes, calculating compound interest, and analyzing exponential growth or decay.
2. How do I calculate logarithms without a calculator?
To calculate logarithms without a calculator, you can use the properties of logarithms and basic algebraic manipulations. For example, if you have the equation log(base b) x = y, you can rewrite it as b^y = x. By using the properties of exponents, you can simplify the equation and solve for the unknown variable. It is also helpful to memorize the logarithmic values of common numbers, such as logarithm of 10 to the base 10 is 1, logarithm of 100 to the base 10 is 2, and so on.
3. What are the different types of logarithms?
There are several types of logarithms commonly used in mathematics. The most common type is the natural logarithm, denoted as ln, which has a base of e (Euler's number, approximately 2.71828). The natural logarithm is often used in calculus and exponential growth problems. Another widely used type is the common logarithm, denoted as log(base 10), which has a base of 10. The common logarithm is frequently used in logarithmic scales and calculations involving powers of 10. Additionally, there are logarithms with other bases, such as log(base 2), log(base e), and log(base 3), which are used in specific applications.
4. Can logarithms be negative?
Logarithms can be negative if the number being evaluated is between 0 and 1. For example, log(base 2) of 0.5 is -1, as 2 raised to the power of -1 equals 0.5. However, logarithms with negative values are not defined for numbers less than or equal to 0, as the logarithm of 0 or a negative number is undefined. It is important to note the domain of the logarithmic function when working with negative numbers.
5. How are logarithms used in real-life applications?
Logarithms have various real-life applications in fields such as finance, science, and engineering. In finance, logarithms are used to calculate compound interest, analyze investment returns, and model financial data. In science, logarithms are used to measure the intensity of earthquakes (Richter scale), calculate pH levels of acids and bases, and analyze exponential growth or decay in biological populations. In engineering, logarithms are used in signal processing, telecommunications, and electrical circuit analysis. Logarithms help simplify complex calculations and provide a useful tool for expressing large or small quantities in a more manageable form.
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