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Logarithm Question

What Are Logarithm Questions?

Logarithm questions might seem tricky at first, but they’re actually just smart ways to deal with really big (or really small) numbers. These questions help you break down tough calculations into simpler steps using special rules called logarithm properties. In most exams - from school tests in Classes 9, 10, and 11 to Maths Olympiads - you’ll find problems that ask you to convert between forms, solve equations, or simplify expressions. The best part? Once you get the hang of the rules, solving logarithm questions can feel like solving a puzzle. With a little practice, you’ll start to notice patterns and develop a better grip on exponential and algebraic problems too.

Table of Contents

Importance of Logarithms in Maths
Logarithm Properties
Logarithm Questions with Answer
Logarithm Practice Questions
Common Logarithms vs Natural Logarithms
Basic Logarithm Rules and Laws
Logarithm Formula and Rules
How to Solve Logarithm Questions
Logarithmic Form vs Exponential Form
Common Mistakes and Tips
Applications of Logarithms in Real Life
Solved Logarithm Questions with Answer
Conclusion
FAQs

Importance of Logarithms in Maths

Logarithms play a crucial role in:

Simplifying multiplication, division, and exponentiation into addition, subtraction, and multiplication using logarithm properties.
Solving exponential equations that are otherwise difficult to handle.
Scaling data in scientific calculations and measurements.
Analyzing exponential growth and decay in fields like biology, chemistry, physics, and finance.
Helping in data visualization, such as plotting data on a logarithmic scale for better interpretation.
Supporting concepts in calculus, algebra, and statistics.
Providing the foundation for more advanced mathematical operations and logarithm questions with answer.

Mastering logarithm questions, logarithm formula, and logarithm properties ensures confidence in handling various mathematical challenges.

Logarithm Properties

Understanding the logarithm properties is essential for solving logarithm questions efficiently. Key logarithm properties include:

Product Property:
log_b (M × N) = log_b M + log_b N

Quotient Property:
log_b (M / N) = log_b M − log_b N

Power Property:
log_b (M^k) = k × log_b M

Change of Base Property:
log_b M = log_k M / log_k b

Zero and One Properties:
log_b (1) = 0
log_b (b) = 1

Logarithm Questions with Answer

Here are some tricky logarithm questions with answers:

1. Express 5³ = 125 in logarithmic form

Solution: log₅(125) = 3

2. Express log₁₀(1) = 0 in exponential form

Solution: 10⁰ = 1

3. Find log₄(32)

Solution:
log₄(32) = x
4ˣ = 32 = 2⁵ = (2²)ˣ
2²ˣ = 2⁵ → 2x = 5 → x = 5/2
Answer: log₄(32) = 5/2

4. Solve: log₅(x – 7) = 1

5¹ = x – 7 → x = 5 + 7 = 12

5. If logₐ(m) = n, express aⁿ⁻¹ in terms of m

From logₐ(m) = n → aⁿ = m
aⁿ⁻¹ = m/a

6. Solve: log(x – 1) + log(x + 1) = log(21)

log[(x – 1)(x + 1)] = log(21)
x² – 1 = 21 → x² = 22 → x = √22
(Reject –√22 as log of negative is undefined)

7. Simplify: log(75/16) – 2log(5/9) + log(32/243)

Using log rules:
= log[(75/16) × (81/25) × (32/243)]
= log(2)
Answer: log₂

8. Convert: 2log(x) + 3log(y) = log(a) to exponential form

log(x²y³) = log(a) → x²y³ = a

9. Prove: 2log(15/18) – log(25/162) + log(4/9) = log(2)

Using log rules:
= log[(15/18)² × (4/9) ÷ (25/162)]
= log(2) ✔

10. Express log₁₀(2) + 1 as a single logarithm

= log₁₀(2) + log₁₀(10) = log₁₀(20)

11. Find x if log₁₀(x – 10) = 1

x – 10 = 10 → x = 20

12. Solve: log(x + 5) + log(x – 5) = 4log(2) + 2log(3)

log(x² – 25) = log(16 × 9) = log(144) → x² – 25 = 144 → x² = 169 → x = 13

13. If (log(225) / log(15)) = log(x), find x

log(225) = log(15²) = 2log(15)
log(x) = 2 → x = 10² = 100

Logarithm Practice Questions

Try solving these logarithm questions on your own:

If log(x) = m + n and log(y) = m – n, find log(10x/y²) in terms of m and n.
Express 3⁻² = 1/9 in logarithmic form.
Convert log(0.01) = –2 into exponential form.
Find log₂₇(1/81)
Solve: log₇(2x² – 1) = 2

Common Logarithm vs Natural Logarithm

Aspect	Common Logarithm	Natural Logarithm
Notation	log₁₀(x) or simply log(x)	ln(x)
Base	10 (Base 10)	e (Euler's number ≈ 2.718)
Example	log(100) = 2 (since 10² = 100)	ln(e³) = 3 (since e³ = e³)
Inverse of	Exponential function: 10ˣ	Exponential function: eˣ
Calculator Key	"log"	"ln"
Used in	Basic math, engineering, common scale values	Higher mathematics, calculus, compound growth
Change of Base Formula	logₐ(b) = log₁₀(b) / log₁₀(a)	logₐ(b) = ln(b) / ln(a)
Conversion	log(x) = ln(x) / ln(10)	ln(x) = log(x) × ln(10)
Differentiation	d/dx[log(x)] = 1 / (x ln(10))	d/dx[ln(x)] = 1 / x
Integration	∫ log(x) dx = x log(x) – x / ln(10) + C	∫ ln(x) dx = x ln(x) – x + C

Basic Logarithm Rules and Laws

Rule / Law	Formula	Description
Product Rule	log<sub>b</sub>(mn) = log<sub>b</sub>(m) + log<sub>b</sub>(n)	Log of a product equals the sum of logs
Quotient Rule	log<sub>b</sub>(m/n) = log<sub>b</sub>(m) – log<sub>b</sub>(n)	Log of a quotient equals the difference of logs
Power Rule	log<sub>b</sub>(mⁿ) = n × log<sub>b</sub>(m)	Power becomes a multiplier
Zero Rule	log<sub>b</sub>(1) = 0	Log of 1 is always 0 regardless of the base
Identity Rule	log<sub>b</sub>(b) = 1	Log of base to itself is 1
Change of Base Rule	log<sub>b</sub>(a) = log<sub>k</sub>(a) / log<sub>k</sub>(b)	Allows conversion between bases using common or natural logs
Log of Reciprocal	log<sub>b</sub>(1/m) = –log<sub>b</sub>(m)	Log of reciprocal is negative of the log
Negative Argument (Not Defined)	log<sub>b</sub>(–x) is undefined	Logarithms are only defined for positive real numbers
Base Condition	b > 0 and b ≠ 1	Base of a logarithm must be positive and not equal to 1

How to Solve Logarithm Questions

To solve logarithm questions, follow these steps:

Rewrite the Equation
Convert the logarithmic equation to exponential form using the logarithm formula.
Apply Properties
Use logarithm properties to simplify expressions.
Isolate the Variable
Solve for the variable using algebraic manipulation.
Check Domain
Ensure solutions make the logarithm argument positive.

Practicing logarithm questions with answer helps reinforce these methods and builds confidence.

Logarithmic Form vs Exponential Form

Aspect	Logarithmic Form	Exponential Form
General Form	log<sub>b</sub>(a) = x	b<sup>x</sup> = a
Base	b	b
Exponent	x (result of the log)	x (power applied to base)
Result	x	a
Example	log<sub>2</sub>(8) = 3	2<sup>3</sup> = 8
Conversion	log<sub>b</sub>(a) = x → b<sup>x</sup> = a	b<sup>x</sup> = a → log<sub>b</sub>(a) = x

Summary of Notations

b = base of the logarithm
m, n, a = positive real numbers
log typically means log base 10 (common log)
ln means log base e (natural log)

Common Mistakes and Tips

While solving logarithm questions, students often make the following errors:

Forgetting that the argument of a logarithm must be positive.
Mixing up logarithm properties, especially product and quotient rules.
Neglecting to convert between logarithmic and exponential forms when necessary.
Forgetting the domain restrictions, leading to extraneous solutions.

Tips:

Always check the domain of your answer.
Memorize key logarithm properties.
Practice logarithm questions with answer for fluency.
Use the logarithm formula carefully for conversions.

Applications of Logarithms in Real Life

Logarithms have vast applications, including:

Sound Intensity (Decibels): Measuring loudness on a logarithmic scale.
pH Calculation: In chemistry, pH = –log[H⁺].
Earthquake Magnitude: Richter scale is logarithmic.
Finance: Modeling compound interest and investment growth.
Data Compression: Logarithmic transformations reduce data size.
Computing: Algorithms like Big O notation often use logarithmic complexity.

Mastering logarithm questions, logarithm properties, and the logarithm formula enables tackling real-life problems efficiently.

Solved Logarithm Questions with Answer

Example 1:
Question: Solve log₁₀(1000).
Solution:
10 raised to what power equals 1000?
10³ = 1000
So, log₁₀(1000) = 3

Example 2:
Question: Evaluate log₂(32).
Solution:
2 raised to what power equals 32?
2⁵ = 32
So, log₂(32) = 5

Example 3:
Question: Find log₅(125).
Solution:
5 raised to what power equals 125?
5³ = 125
So, log₅(125) = 3

Example 4:
Question: Simplify log₁₀(5 × 20).
Solution:
log₁₀(5 × 20) = log₁₀(5) + log₁₀(20)
log₁₀(5) ≈ 0.699
log₁₀(20) ≈ 1.301
Sum = 0.699 + 1.301 = 2
So, log₁₀(100) = 2

Example 5:
Question: Calculate log₃(81).
Solution:
3 raised to what power equals 81?
3⁴ = 81
So, log₃(81) = 4

Example 6:
Question: Find log₁₀(0.01).
Solution:
10 raised to what power equals 0.01?
10⁻² = 0.01
So, log₁₀(0.01) = -2

Example 7:
Question: Simplify log₁₀(1).
Solution:
10 raised to what power equals 1?
10⁰ = 1
So, log₁₀(1) = 0

Conclusion

Mastering logarithm questions is essential for excelling in mathematics and numerous real-life fields. By practicing logarithm questions with answer, students develop confidence in handling diverse problems. Use logarithm properties and the logarithm formula regularly to simplify calculations and deepen your understanding.

Practicing logarithm questions ensures fluency in solving exponential equations, data analysis, and advanced mathematical topics.

Related Links

Value of log : Unlock the secrets behind the value of log. Click here to dive into the full blog and master logarithms easily!

Frequently Asked Questions on Logarithm Questions

1: What is a logarithm, for example?

A logarithm is the exponent to which the base must be raised to get a specific number. For example, log⁡101000=3\log_{10} 1000 = 3 because 103=100010^3 = 1000.