Logarithms are a crucial topic in Class 11 mathematics and hold significant weight in the JEE syllabus. Understanding logarithmic functions, their properties, and how to manipulate logarithmic expressions is essential for solving a wide variety of problems in algebra. Students preparing for JEE must develop strong problem-solving skills related to logarithms, as questions often appear in both straightforward and complex formats. A thorough grasp of this chapter not only helps in JEE but also builds a solid foundation for future mathematical concepts like calculus and exponential growth.
What is a logarithm?
A logarithm is the inverse of an exponential function. In simple terms, ifax= b, thenlogab = x. This means that the logarithm answers the question to what power must the base be raised to obtain a certain number?
Basic questions
- What is the value of log28?
- If log5x = 3, what is the value of x?
- Simplify log101000
What are the fundamental laws of logarithms?
There are three main laws of logarithms that are frequently used in problem solving
- Product Ruleloga(mn) = logam + logan
- Quotient Ruleloga(m/n) = logam − logan
- Power Ruleloga(mn) = n·logam
Application questions
- Simplify log327 + log39
- Evaluate log2(32/4)
- Solve for x log5(x2) = 6
How do we change the base of a logarithm?
To convert a logarithm from one base to another, the change of base formula is used
logab = logcb / logca
This is particularly useful when solving logarithmic expressions where the base is not standard or needs to be simplified for computation.
Example questions
- Evaluate log210 using base 10 logarithms.
- Simplify log416 using base 2.
- If loga2 = 0.3010 and loga3 = 0.4771, find loga6.
How do logarithms help solve exponential equations?
Logarithms are used to solve equations where the variable is in the exponent. By applying logarithms on both sides, we can bring down the exponent and simplify the equation.
Problem-solving questions
- Solve for x 2x= 32
- Solve for x 5x+1= 125
- Solve 102x= 1000
What are common mistakes in logarithmic problems?
Many students make errors while solving logarithmic questions due to misapplication of laws or ignoring domain restrictions. The domain of a logarithmic function is strictly positive, meaning the argument of the log must be greater than zero.
Conceptual questions
- Why is log10(−5) undefined?
- What is the domain of the function f(x) = log(x − 2)?
- Can loga0 ever be defined for any base a?
How to graph logarithmic functions?
Understanding the graphical behavior of logarithmic functions helps in visualizing their properties. The graph of y = logax is increasing when a >1 and decreasing when 0 < a < 1. It always passes through the point (1, 0) and has a vertical asymptote at x = 0.
Graph-related questions
- Sketch the graph of y = log2x.
- Describe the transformation of y = log3(x − 1).
- What is the asymptote of y = log10x?
What is the significance of logarithmic identities in JEE?
In JEE, logarithmic identities are often hidden within complex algebraic expressions. Recognizing and applying them quickly is key to solving problems efficiently. This includes using identities in reverse, such as writing a sum of logs as a single logarithm.
JEE-style mixed questions
- If log10(x + y) = 1 and log10x + log10y = 1, find x and y.
- Simplify log28 + log24 − log216
- If logax = m and logay = n, express loga(x²y³) in terms of m and n.
What are logarithmic inequalities?
Logarithmic inequalities involve solving inequalities where the logarithm appears. These require understanding the domain and monotonic nature of logarithmic functions. For example, if a >1, then logax is an increasing function.
Inequality questions
- Solve log2(x + 3) > 2
- Find the solution set log5(x − 1) ≤ 1
- Determine x for which log(x) < log(x + 1)
How do logarithms appear in real-world applications?
Logarithms are not just theoretical; they appear in many real-life scenarios like measuring earthquake intensity (Richter scale), sound level (decibels), pH in chemistry, and population growth models. Questions in JEE may include contexts that relate logarithmic understanding to practical examples.
Application-based questions
- If pH = −log[H+], find the pH when [H+] = 10−3
- The loudness of a sound is given by L = 10·log(I/I₀). If I = 1000I₀, find L.
- Earthquake magnitude M = log(E/E₀). If E = 10000E₀, calculate M.
Questions on logarithms in Class 11 JEE preparation span from simple definitions and laws to complex problems involving equations and inequalities. A strong command of logarithmic properties, change of base, domain understanding, and application in solving exponential equations is essential for success. By practicing a wide range of conceptual and numerical questions, students can gain confidence and accuracy in this topic. Given its importance across math and science, mastering logarithms serves as a powerful tool not only for JEE but also for future academic and practical challenges.