How to Solve Logarithm Questions Easily for Exams
The concept of logarithm questions with solutions plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Solving logarithm questions accurately and quickly is essential for students preparing for board exams, JEE, Olympiads, and other competitive tests. By practicing various log problems and understanding stepwise solutions, you can master this topic and improve your calculation speed.
What Is Logarithm Questions with Solutions?
A logarithm is defined as the exponent by which a specified base must be raised to obtain a given number. In simple terms, log questions ask, “To what power must a base (like 10 or e) be raised to yield another number?” You’ll find this concept applied in areas such as exponential growth, pH calculations, sound intensity, and computer algorithms.
Key Formula for Logarithm Questions with Solutions
Here’s the standard formula: \( \log_b a = x \) means \( b^x = a \)
Common bases: 10 (common logarithm) and e (natural logarithm, written as ln)
Logarithm Laws to Remember
| Law | Formula | Example |
|---|---|---|
| Product Rule | \( \log_b(mn) = \log_b m + \log_b n \) | \( \log_{10}(2 \times 5) = \log_{10}2 + \log_{10}5 \) |
| Quotient Rule | \( \log_b\left(\frac{m}{n}\right) = \log_b m - \log_b n \) | \( \log_2\frac{8}{2} = \log_2 8 - \log_2 2 \) |
| Power Rule | \( \log_b m^k = k \log_b m \) | \( \log_{10} 100^2 = 2 \log_{10} 100 \) |
| Change of Base Rule | \( \log_b a = \frac{\log_k a}{\log_k b} \) | \( \log_2 8 = \frac{\log_{10}8}{\log_{10}2} \) |
Step-by-Step Illustration: Sample Logarithm Questions with Solutions (Class 11/12/Competitive)
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Solve for x: \( \log_{10} x = 2 \)
1. Write in exponential form: \( 10^2 = x \ )
2. Compute: \( x = 100 \ )
Final Answer: x = 100
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Solve \( \log_{2} (x) + \log_{2} (x-2) = 3 \)
1. Use product rule: \( \log_{2}[x(x-2)] = 3 \)
2. Exponential form: \( x(x-2) = 2^3 = 8 \)
3. Expand: \( x^2 - 2x = 8 \) ⇒ \( x^2 - 2x - 8 = 0 \)
4. Factor: \( (x-4)(x+2) = 0 \)
5. Solutions: \( x = 4 \) or \( x = -2 \)
6. Since x must be positive for logs, \( x = 4 \)
Final Answer: x = 4
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Simplify: \( \log_5 25 + \log_2 8 \)
1. \( \log_5 25 = 2 \) (since \( 5^2 = 25 \))
2. \( \log_2 8 = 3 \) (since \( 2^3 = 8 \))
3. Add: \( 2 + 3 = 5 \)
Final Answer: 5
Speed Trick or Vedic Shortcut
Here’s a quick shortcut: To solve logs of numbers like \( \log_{10} (2 \times 5) \), just split into \( \log_{10}2 + \log_{10}5 \). Also, for base change:
\( \log_a b = \dfrac{\log_{10} b}{\log_{10} a} \).
These help you calculate logs fast without tables.
Example Trick: \( \log_2 32 \)?
Count the number of times 2 is multiplied to get 32 (2 × 2 × 2 × 2 × 2 = 32, so 5 times):
Answer: 5
Tricks like this are practical in exams like JEE, NDA, and state boards. Vedantu’s maths classes share more such tips to make calculations easy and fast.
Try These Yourself
- Find the value of x if \( \log_3 x = 4 \)
- Simplify \( \log_{10} 1000 - \log_{10} 10 \)
- Solve \( \log_x 16 = 2 \)
- If \( \log_{a} b = 0.5 \), find b in terms of a.
- If \( x = \log_{2} (y) \) and \( y = 8 \), what is x?
Frequent Errors and Misunderstandings
- Using log rules for sums or differences inside the log, e.g., \( \log(a + b) \neq \log a + \log b \).
- Calculating log of negative or zero (undefined in real numbers).
- Forgetting to check final values are positive and valid for the log base.
Relation to Other Concepts
The idea of logarithm questions with solutions connects closely with exponents, index laws, and algebraic manipulation. Mastering logarithms makes it easier to solve exponential growth/decay, compound interest, and function problems in higher maths.
Classroom Tip
A quick way to remember log rules is to write the product, quotient, and power rules in a triangle memory chart. Vedantu teachers often show this visually during live revision sessions.
We explored logarithm questions with solutions—from definition, formulas, worked examples, common mistakes, and links to other maths topics. Continue practicing with Vedantu’s resources and interactive solutions to become confident in tackling any logarithm problem, whether it’s for school, boards, or competitive exams.
Related Math Links
FAQs on Logarithm Questions with Solutions and Practice PDFs
1. What is a logarithm in Maths?
A logarithm is the inverse operation of exponentiation. It answers the question: "To what power must we raise a base to get a certain number?" For example, in the equation 102 = 100, the logarithm (base 10) of 100 is 2, written as log10(100) = 2. Understanding logarithms is crucial for solving many mathematical problems, particularly those involving exponential growth and decay.
2. How do I quickly solve logarithm questions in exams?
Mastering logarithm properties is key. Familiarize yourself with the product rule (logb(xy) = logb(x) + logb(y)), quotient rule (logb(x/y) = logb(x) - logb(y)), and power rule (logb(xy) = y logb(x)). Practice applying these rules to simplify complex expressions. Knowing how to change the base of a logarithm can also be helpful. Regularly solve practice problems to improve speed and accuracy.
3. What are the main properties of logarithms?
The fundamental properties of logarithms include:
• **Product Rule:** logb(xy) = logbx + logby
• **Quotient Rule:** logb(x/y) = logbx - logby
• **Power Rule:** logb(xn) = n logbx
• **Change of Base Rule:** logbx = (logax)/(logab)
Understanding these properties allows for simplification and efficient solving of logarithmic equations.
4. Do competitive exams like JEE ask direct log questions?
Yes, the JEE Main and Advanced exams frequently include questions involving logarithms, often integrated within calculus, algebra, or other mathematical concepts. These questions test your understanding of logarithm properties and your ability to apply them in complex problem-solving scenarios. Expect to encounter both direct and indirect applications of logarithmic concepts.
5. Where can I download logarithm questions with solved answers for practice?
Vedantu provides a wealth of resources, including downloadable PDFs containing logarithm questions with solutions, tailored for various levels (Class 10, 11, 12, and competitive exams). These resources are designed to aid your exam preparation and build your confidence in tackling logarithm-related problems. Access these materials through the Vedantu website or app.
6. What are the common mistakes students make when solving log questions?
Common errors include misapplying logarithm properties, incorrect handling of negative numbers or zero within logarithms (remember, the argument must be positive), and failing to properly simplify expressions before attempting solutions. Careful attention to detail and consistent practice are crucial to avoid these mistakes.
7. How to identify ‘hidden’ log mistakes in MCQ options?
Carefully check each MCQ option by substituting the given values and applying the logarithm properties. Look for options that result in a mathematically impossible outcome, like taking the logarithm of a non-positive number. Look for slight differences in expressions that hide errors in the simplification or application of logarithm rules.
8. Is there a shortcut to convert any log base to base 10 or *e* for calculators?
Yes, the change of base formula allows this. To convert logbx to base 10, use: log10x / log10b. Similarly, for base *e*, use: ln x / ln b. This enables efficient calculation using standard calculator functions.
9. How do logarithms help in solving real-life growth or decay (physics/finance) problems?
Logarithms are essential for modeling exponential growth and decay in various fields. In physics, they describe radioactive decay, while in finance, they are used in compound interest calculations. The logarithmic scale simplifies the analysis of large ranges of values, making it easier to visualize and understand growth or decay patterns over time.
10. What’s the difference between principal value and general value in log equations?
The principal value of a logarithm is the single value within a specific range (often 0 to 2π for trigonometric functions). The general value includes all possible solutions, which can be represented using a general formula incorporating the periodicity of the logarithm function.
11. How can I master log properties for faster problem-solving?
Consistent practice is crucial. Begin with basic problems, gradually increasing the complexity. Use flashcards or create a summary chart of the logarithm properties to aid memorization. Focus on understanding the underlying principles rather than rote memorization; this will help you apply them effectively in various scenarios.
12. What resources are available on Vedantu to help me with logarithms?
Vedantu offers various resources including solved examples, practice problems, and downloadable PDFs. Expert-led online sessions provide additional support and clarification. The platform aims to provide a comprehensive learning experience to help you master logarithms and related concepts.
