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Mathematics Grade IX
Mathematics Grade 9
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Ch-2 Logarithms
School Revise · Grade 9 · Advanced Maths (Optional) · Chapter 2
Logarithms
Mathematics at Advanced Level, Class 9. A logarithm answers the question, what power gives this number. This optional chapter defines the logarithm, gives its laws that turn multiplication into addition, and uses common logarithms.
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What this chapter is about
Multiplying large numbers is hard, adding is easy. Logarithms turn one into the other. In this chapter we define a logarithm as the power that produces a number, learn the laws that follow, and use base ten common logarithms.
1. What a logarithm is
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Definition of a logarithm log to base b of a equals x means exactly that b raised to the power x equals a: log₺a = x ⇔ bₓ = a A logarithm just names the power. Since 2³ = 8, we say log₂8 = 3. Two useful facts follow at once: log₺1 = 0, because b⁰ = 1, and log₺b = 1, because b¹ = b. In real life: Earthquake strength, sound loudness in decibels and acidity as pH are all measured on logarithmic scales, so each step is a multiplying jump. |
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Worked example 1 Question. Find log₂8.
Answer: log₂8 = 3. |
2. The laws of logarithms
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The three log laws log(xy) = log x + log y; log(x/y) = log x − log y; log(xⁿ) = n log x The first turns a product into a sum, the second turns a division into a subtraction, and the third brings a power down to the front as a multiplier. These are what make logarithms so useful for calculation. In real life: Before calculators, engineers multiplied huge numbers quickly by adding their logarithms, using printed log tables and slide rules. |
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Worked example 2 Question. Use the laws to find log₁₀(100 × 1000).
Answer: log₁₀(100 × 1000) = 5. |
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Worked example 3 Question. Solve log₂x = 4.
Answer: x = 16. |
Practise with the interactive
Try this chapter hands on: change the values and watch the answer update live and animate. The interactive opens right here in the lesson.
Practice set A, multiple choice
1. log₺1 is always …
Zero, because any base to the power 0 equals 1, so the power that gives 1 is 0.
2. log(xy) equals …
log x + log y. A logarithm turns a product into a sum.
3. log₃81 equals …
| 1 | What power of 3 gives 81?. 3⁴ = 81. |
| 2 | So. log₃81 = 4. |
4. Bringing a power to the front uses the law …
log(xⁿ) = n log x, which turns a power into a multiplier.
Practice set B, short answer
1. Find log₁₀10000.
| 1 | What power of 10 gives 10000?. 10⁴ = 10000. |
| 2 | So. = 4. |
2. Write log₂(x/y) using the log laws.
log₂x − log₂y, because a division inside a logarithm becomes a subtraction outside it.
3. Solve log₃x = 3.
| 1 | Rewrite in power form. x = 3³. |
| 2 | Work it out. = 27. |
Quick summary
| Idea | The idea |
| Logarithm | The power that gives a number. |
| Definition | log₺a = x ⇔ bₓ = a. |
| log 1 | = 0. |
| Product law | log(xy) = log x + log y. |
| Quotient law | log(x/y) = log x − log y. |
| Power law | log(xⁿ) = n log x. |
| Open the Virtual Lab |
These free Grade 9 Advanced Maths notes explain logarithms, the log laws, common logarithms and solving log equations, with clear step by step worked examples and labelled diagrams for every student using the optional Advanced Level book.
© 2026 School Revise. All rights reserved. This lesson is original content written by School Revise, aligned to the CBSE Class 9 Mathematics at Advanced Level (Optional) syllabus. Unauthorised copying, reproduction or redistribution is not permitted. Curriculum names are used only to indicate alignment.
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