Disclaimer: This content is based on questions from past papers from the International Baccalaureate (IB). All questions are sourced from publicly available IB materials. This website and article are not affiliated with or endorsed by the International Baccalaureate Organization (IBO). All rights to the original questions are owned by IBO.
Problem 2: Mastering Logarithms! [Total Marks: 5]
Hey students! Ready to dive into some logarithm fun? This problem is a great chance to sharpen your skills and impress your math teacher. Let’s get started!
You’re told that:
log₁₀(a) = 1/3
where a > 0. Your mission is to solve these two parts:
a) Find the value of log₁₀(1/a). [2 marks]
b) Calculate the value of log₁₀₀₀(a). [3 marks]
Solution to Problem 2: Let’s Crack It!
Solution to Part (a): Flipping the Fraction
To find log₁₀(1/a), let’s use a super handy logarithm rule: logₓ(1/y) = -logₓ(y). This means:
log₁₀(1/a) = -log₁₀(a)
We know log₁₀(a) = 1/3, so:
log₁₀(1/a) = -1/3
Final Answer for (a): -1/3
Great job! You just used a logarithm property like a pro!
Solution to Part (b): Changing the Base
Now, let’s tackle log₁₀₀₀(a). This one’s a bit trickier, but you’ve got this! We’ll use the change of base formula: logₓ(y) = logₖ(y) / logₖ(x). Let’s pick base 10 since it’s familiar:
log₁₀₀₀(a) = log₁₀(a) / log₁₀(1000)
Since 1000 = 10³, we know log₁₀(1000) = log₁₀(10³) = 3. And we’re given log₁₀(a) = 1/3. Plugging in:
log₁₀₀₀(a) = (1/3) / 3 = 1/9
Final Answer for (b): 1/9
Awesome work! You just handled a base change like a math superstar!
Alternative Ways to Solve: Think Like a Detective!
Sometimes, there’s more than one way to solve a problem. Let’s explore some alternative approaches to make sure you’re ready for any logarithm challenge.
Alternative Solution to Part (a)
Notice that 1/a = a⁻¹. There’s a logarithm rule that says logₓ(yⁿ) = n·logₓ(y). So:
log₁₀(1/a) = log₁₀(a⁻¹) = -1·log₁₀(a) = -log₁₀(a)
Since log₁₀(a) = 1/3:
log₁₀(1/a) = -1/3
Final Answer for (a): -1/3
See? Same answer, different path. You’re building a toolbox of math tricks!
Alternative Solution to Part (b)
Let’s try expressing ‘a’ first. Since log₁₀(a) = 1/3, we can write:
a = 10^(1/3)
Now, we need log₁₀₀₀(a) = log₁₀₀₀(10^(1/3)). Using the change of base formula:
log₁₀₀₀(10^(1/3)) = log₁₀(10^(1/3)) / log₁₀(1000)
We know log₁₀(10^(1/3)) = 1/3 and log₁₀(1000) = 3, so:
log₁₀₀₀(10^(1/3)) = (1/3) / 3 = 1/9
Final Answer for (b): 1/9
Two methods, same result! You’re unstoppable!
Pro Tips for Logarithm Problems
Want to ace logarithm questions every time? Here’s your game plan:
- Know Your Rules: Memorize key logarithm properties like log(1/x) = -log(x) or log(xⁿ) = n·log(x).
- Master Base Changes: Use logₓ(y) = logₖ(y) / logₖ(x) when bases don’t match.
- Rewrite Logs: If you’re stuck, try expressing logs in exponential form (e.g., logₓ(a) = c means a = x^c).
- Double-Check Math: Be careful with fractions and exponents—small mistakes can trip you up!
- Practice, Practice, Practice: The more you solve, the more confident you’ll get.
Keep these tips in your back pocket, and you’ll be solving logarithms like a champ!
How Your Work Will Be Graded
Here’s what your teacher is looking for:
Part (a): [2 marks]
- 1 mark for using a logarithm property correctly (e.g., log₁₀(1/a) = -log₁₀(a)).
- 1 mark for getting the correct answer: -1/3.
Part (b): [3 marks]
- 1 mark for using the change of base formula or an equivalent method.
- 1 mark for correctly finding log₁₀(1000) = 3.
- 1 mark for the final answer: 1/9.
Total: 5 marks
Nail these steps, and you’re on your way to a perfect score!
Watch Out for These Common Mistakes!
Don’t fall into these traps—here’s how to avoid them:
- Mistake: Thinking log(1/a) = 1/log(a).
Fix: Remember log(1/a) = -log(a). Practice the property! - Mistake: Assuming log₁₀₀₀(a) = log₁₀(a).
Fix: Use the change of base formula to handle different bases. - Mistake: Messing up the math, like saying (1/3) ÷ 3 = 1/3.
Fix: Double-check your fractions: (1/3) ÷ 3 = 1/9. - Mistake: Forgetting to write the base (e.g., log(a) instead of log₁₀(a)).
Fix: Always specify the base to avoid confusion.
Stay sharp, and you’ll dodge these pitfalls!
Practice Problems: Test Your Skills!
Ready to level up? Try these problems to solidify your logarithm skills.
Practice Problem 1: Inverse Logarithm [2 marks]
Given log₁₀(b) = 1/4, find log₁₀(1/b).
Solution:
log₁₀(1/b) = -log₁₀(b) = -1/4
Answer: -1/4
Practice Problem 2: Change of Base [3 marks]
Using log₁₀(b) = 1/4, calculate log₁₀₀(b).
Solution:
log₁₀₀(b) = log₁₀(b) / log₁₀(100)
log₁₀(b) = 1/4, and log₁₀(100) = 2, so:
log₁₀₀(b) = (1/4) / 2 = 1/8
Answer: 1/8
Advanced Challenges: Push Your Limits!
Feeling confident? These advanced problems will stretch your brain!
Advanced Problem 1: Combined Logarithms [5 marks]
Given log₁₀(c) = 2/5, find log₁₀(c²/√c) and log₁₀₀₀₀(c).
Solution:
For log₁₀(c²/√c):
c²/√c = c² · c^(-1/2) = c^(2 – 1/2) = c^(3/2)
log₁₀(c^(3/2)) = (3/2)·log₁₀(c) = (3/2)·(2/5) = 3/5
For log₁₀₀₀₀(c):
log₁₀₀₀₀(c) = log₁₀(c) / log₁₀(10000)
log₁₀(10000) = log₁₀(10⁴) = 4, so:
log₁₀₀₀₀(c) = (2/5) / 4 = 2/20 = 1/10
Answer: 3/5, 1/10
Advanced Problem 2: Logarithmic Equation [4 marks]
Given log₁₀(d) = 1/2, solve for x in log₁₀₀(x) = log₁₀₀₀(d).
Solution:
First, find log₁₀₀₀(d):
log₁₀₀₀(d) = log₁₀(d) / log₁₀(1000) = (1/2) / 3 = 1/6
So, log₁₀₀(x) = 1/6. Since log₁₀₀(x) = log₁₀(x) / log₁₀(100) and log₁₀(100) = 2:
log₁₀(x) / 2 = 1/6
log₁₀(x) = 2/6 = 1/3
x = 10^(1/3)
Answer: 10^(1/3)
Keep Going, Math Rockstar!
Logarithms might seem tricky at first, but with practice, you’ll be solving them in your sleep. Use the properties, stay careful with your calculations, and don’t be afraid to try different approaches. You’ve got this! Drop a comment on the blog if you have questions or want more practice problems. Happy math-ing!
🙋♂️ Need Help with IB Math?
I offer:
- 1-on-1 doubt-solving sessions
- Personalized mentorship and study plans
- Step-by-step help in IB, IGCSE, A-Level, AP, and Olympiad Mathematics
🌐 Apply for Mentorship or Tutoring Session: https://forms.gle/dL4M14SWSsMC3gfC7
Rishabh Kumar, Elite Private Mentor, Founder, Mathematics Elevate Academy
📥 Download the full solution with detailed steps (PDF): LaTeX formatted (on Request)
💬 Enjoyed the solution? Drop a comment below — I’d love to hear your thoughts or questions!
🎓 Join my Discord channel for LaTeX-formatted PDF solutions, exclusive math resources, and student discussions: https://discord.gg/vXAAkxrT
📢 Follow me for more:
- 🔗 Medium: https://medium.com/@mathematicselevateacademy001
- 💼 LinkedIn: https://www.linkedin.com/in/rishabh-kumar-iitg-isi/
- 📱 Telegram (optional): t.me/mea_global
