Logarithm Question

 

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 many problem-solving situations, you’ll come across tasks that involve converting between forms, solving equations, or simplifying 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

• 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
• Common Mistakes and Tips
• Conclusion
• FAQs on Logarithm Question

 

Logarithm Properties

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

Product Property:
logb(M×N)=logbM+logbN

Quotient Property:
logb(M/N)=logbM−logbN

 Power Property:
logb(Mk)=k×logbM

 Change of Base Property:

logbM=logkM/logkb

Zero and One Properties:
logb(1)=0

logb(b)=1

 

Logarithm Questions with Answer

1. Express 2^4 = 16 in logarithm form.

Solution:
We know, a^b = c ⇒ log_a c = b

Here, 2^4 = 16.
So, log₂16 = 4.

2. Express log₁₀1000 = 3 in exponential form.

Solution:
We know, log_a c = b ⇒ a^b = c

So, log₁₀1000 = 3 ⇒ 10³ = 1000.

3. Find the log of 81 to the base 3.

Solution:
log₃81 = x

⇒ 3^x = 81

But 81 = 3^4.
So, 3^x = 3^4 ⇒ x = 4.

∴ log₃81 = 4.

4. Find x if log₂(x - 3) = 4.

Solution:
log₂(x - 3) = 4

⇒ 2^4 = x - 3

⇒ 16 = x - 3

⇒ x = 19.

5. If log_a m = n, express a^(n+1) in terms of a and m.

Solution:
log_a m = n ⇒ a^n = m.

Now,
a^(n+1) = a^n × a = m × a.

So, a^(n+1) = ma.

6. Solve for x: log(x+2) + log(x-2) = log 21.

Solution:
log(x+2) + log(x-2) = log 21

⇒ log[(x+2)(x-2)] = log 21

⇒ log(x² - 4) = log 21

⇒ x² - 4 = 21

⇒ x² = 25

⇒ x = ±5

Since log of a negative number is not defined,
∴ x = 5.

7. Express log(48/25) - 2log(4/5) + log(27/32) in terms of log 2 and log 3.

Solution:
log(48/25) - 2log(4/5) + log(27/32)

= log(48/25) - log((4/5)²) + log(27/32)

= log(48/25) - log(16/25) + log(27/32)

= log((48/25)/(16/25)) + log(27/32)

= log(48/16) + log(27/32)

= log 3 + log(27/32)

= log(3 × 27/32)

= log(81/32)

= 4log 3 - 5log 2.

8. Express 3log x + 2log y = log a in non-logarithmic form.

Solution:
3log x + 2log y = log a

= log(x³) + log(y²)

= log(x³y²)

So, x³y² = a.

9. Prove: 2log(12/18) - log(16/81) + log(3/2) = log 2.

Solution:
LHS = 2log(12/18) - log(16/81) + log(3/2)

= log(12/18)² - log(16/81) + log(3/2)

= log(144/324) - log(16/81) + log(3/2)

= log(4/9) - log(16/81) + log(3/2)

= log((4/9)/(16/81)) + log(3/2)

= log(324/144) + log(3/2)

= log(9/4) + log(3/2)

= log(27/8)

= log 2 .

10. Find x if log(x+6) + log(x-6) = 2log 5.

Solution:
log(x+6) + log(x-6) = 2log 5

⇒ log[(x+6)(x-6)] = log 25

⇒ log(x² - 36) = log 25

⇒ x² - 36 = 25

⇒ x² = 61

⇒ x = ±√61

Since log requires positive values,
∴ x = √61.

 

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

How to Solve Logarithm Questions

To solve logarithm questions, follow these steps:

1. Rewrite the Equation
   Convert the logarithmic equation to exponential form using the logarithm formula.

2. Apply Properties
   Use logarithm properties to simplify expressions.

3. Isolate the Variable
   Solve for the variable using algebraic manipulation.

4. Check Domain
   Ensure solutions make the logarithm argument positive.
   
   

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

 

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.

 

 

 

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.

 

Frequently Asked Questions on Logarithm Questions

1. What is a logarithm, for example?

Ans. A logarithm is the exponent to which the base must be raised to get a specific number. For example, log⁡101000=3\log101000 = 3 because 103=100010^3 = 1000.

 

2. What is the value of log⁡10100\log_{10} 100?

Ans. Since 102=10010^2 = 100, the value of log⁡10100\log10100 is 2.

 

3. How do logarithm properties help in solving problems?

Ans. They simplify complex calculations by turning multiplication into addition, division into subtraction, and exponents into multiplication.

 

4. Can logarithms be negative?

Ans. The value of a logarithm can be negative, but the argument (inside the log) must be positive.

 

5. How do you change the base of a logarithm?

Ans. Change of Base Formula:

logb⁡x=logk⁡xlogk⁡x

 

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