Introduction
Logarithms, often referred to as “logs”, is an integral part of advanced mathematics. Introduced in the early 17th century by John Napier, they are the reverse of exponentiation. Logarithms are extensively used in various fields including computer science, physics, engineering, and statistics. They play a crucial role in calculations involving exponential growth or decay, making them an essential concept to understand for many competitive exams.
Importance in SSC, RRB, and Banking Exams
In SSC, RRB, and banking exams, questions from logarithms usually appear in the quantitative aptitude or mathematics section. While these questions might seem challenging at first, understanding the basic properties of logarithms and practicing problems can make this topic more approachable and manageable.
Basic Concepts and Definitions
A logarithm is the exponent to which the base must be raised to get the number. If b^n = x, then the base ‘b’ logarithm of ‘x’ is ‘n’, which can be written as:
logb(x) = n
Here ‘b’ is the base of the logarithm, ‘x’ is the argument of the logarithm, and ‘n’ is the value of the logarithm.
Basic Properties and Rules of Logarithms
Here are the basic properties of logarithms:
- Product Rule: logb(mn) = logb(m) + logb(n)
- Quotient Rule: logb(m/n) = logb(m) – logb(n)
- Power Rule: logb(m^n) = n*logb(m)
- Identity Rule: logb(b) = 1 and logb(1) = 0
- Change of Base Rule: logb(a) = logc(a) / logc(b), where ‘c’ is the new base.
Advanced Concepts in Logarithms
The advanced concepts of logarithms include:
- Common Logarithms: When the base of a logarithm is 10, it is known as a common logarithm. It is usually written as log(x), without mentioning the base.
- Natural Logarithms: Logarithms with the base ‘e’ (approximately 2.718) are called natural logarithms and are denoted by ln(x).
- Solving Logarithmic Equations: These equations involve variables in the argument or the base of a logarithmic expression. They often require the use of the properties of logarithms to simplify and solve.
Problem-Solving Techniques and Examples
Now, let’s go through a few examples:
Example 1: Solve the logarithmic equation: log2(x) + log2(x – 2) = 3.
Using the product rule of logarithms, the equation can be rewritten as log2(x(x – 2)) = 3, which further simplifies to 2^3 = x^2 – 2x. This equation simplifies to x^2 – 2x – 8 = 0, which factors to (x – 4)(x + 2) = 0, yielding the solutions x = 4 and x = -2. However, a logarithm’s argument must always be positive, so the only valid solution is x = 4.
Example 2: Evaluate log4(64).
Using the definition of a logarithm, we can write 4^n = 64. Since 4^3 = 64, the value of n (which is log4(64)) is 3.
Practice Problems
Try solving these problems to test your understanding:
- Solve for x: log3(x^2) = 4.
- Evaluate: log5(125).
- If log10(x) = 2 and log10(y) = 3, evaluate: log10(100xy^2).
Summary
Logarithms are an important mathematical concept with numerous applications in various fields. A good understanding of the properties and rules of logarithms can simplify many mathematical tasks and help you solve problems more efficiently. With regular practice, you can become proficient in solving logarithm-related problems, enhancing your performance in the quantitative section of competitive exams.