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.
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