Logarithm Questions with Answers Formulas Properties and Step by Step Methods
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 Detailed Solutions and Concepts
1. What is a logarithm in Maths?
A logarithm is the power to which a base must be raised to produce a given number. In simple terms, if by = x, then logbx = y.
- b = base (b > 0, b ≠ 1)
- x = argument (x > 0)
- y = exponent
2. What is the formula for logarithms?
The main logarithm formula is logbx = y ⇔ by = x. Important logarithmic laws include:
- Product rule: logb(xy) = logbx + logby
- Quotient rule: logb(x/y) = logbx − logby
- Power rule: logb(xn) = n logbx
3. How do you solve a logarithmic equation?
To solve a logarithmic equation, rewrite it in exponential form or combine logs using logarithm laws. Steps:
- Isolate the logarithm expression.
- Use log rules to simplify if needed.
- Convert to exponential form.
- Solve the resulting equation and check domain (x > 0).
4. What is the difference between natural log and common log?
The difference is the base: natural log (ln) has base e, while common log (log) has base 10.
- ln x = logex, where e ≈ 2.718
- log x = log10x
5. What are the basic properties of logarithms?
The basic properties of logarithms are the product, quotient, and power rules. Key properties include:
- logb1 = 0
- logbb = 1
- logb(xy) = logbx + logby
- logb(x/y) = logbx − logby
- logb(xn) = n logbx
6. How do you change the base of a logarithm?
You change the base of a logarithm using the change of base formula: logbx = log x / log b. This formula allows evaluation using a calculator.
- Example: log₂8 = log 8 / log 2
- Since log 8 ≈ 0.903 and log 2 ≈ 0.301, result ≈ 3
7. What is the domain of a logarithmic function?
The domain of a logarithmic function is all positive real numbers, meaning x > 0. A logarithm is undefined for zero or negative values.
- For f(x) = log x, domain is (0, ∞)
- For f(x) = log(x − 3), domain is x > 3
8. How do you simplify logarithmic expressions?
To simplify logarithmic expressions, apply logarithm laws such as product, quotient, and power rules. Steps:
- Expand powers using the power rule.
- Break products into sums.
- Break quotients into differences.
9. What are logarithms used for in real life?
Logarithms are used to model exponential growth and decay in real life. Common applications include:
- Compound interest calculations
- Population growth models
- pH scale in chemistry
- Richter scale for earthquakes
10. What are common mistakes when solving logarithm questions?
Common mistakes in logarithm questions include ignoring domain restrictions and misusing log rules. Frequent errors:
- Forgetting that x must be positive
- Incorrectly applying log(x + y) = log x + log y (this is false)
- Not checking extraneous solutions
