What are Common Logarithms?
Common logarithms are logarithms with base 10. They are written as \(\log x\) (without writing the base 10).
Why Base 10?
- Our number system is base 10 (decimal system)
- Powers of 10 are easy to work with: \(10^0=1\), \(10^1=10\), \(10^2=100\), etc.
- Common logs are used in scientific calculations
Properties of Common Logarithms:
| Number | As Power of 10 | Common Log |
|---|---|---|
| 10 | \(10^1\) | \(\log 10 = 1\) |
| 100 | \(10^2\) | \(\log 100 = 2\) |
| 1000 | \(10^3\) | \(\log 1000 = 3\) |
| 0.1 | \(10^{-1}\) | \(\log 0.1 = -1\) |
| 0.01 | \(10^{-2}\) | \(\log 0.01 = -2\) |
Logarithms of Numbers Between 1 and 10:
- \(\log 2 \approx 0.3010\)
- \(\log 3 \approx 0.4771\)
- \(\log 4 = 2 \log 2 \approx 0.6021\)
- \(\log 5 \approx 0.6990\)
- \(\log 6 = \log 2 + \log 3 \approx 0.7781\)
- \(\log 7 \approx 0.8451\)
- \(\log 8 = 3 \log 2 \approx 0.9031\)
- \(\log 9 = 2 \log 3 \approx 0.9542\)
Real-Life Applications:
| Application | What It Measures | Formula |
|---|---|---|
| **Richter Scale** | Earthquake magnitude | \(M = \log(\frac{A}{A_0})\) |
| **Decibels** | Sound intensity | \(L = 10 \log(\frac{I}{I_0})\) |
| **Astronomy** | Star brightness | \(m = -2.5 \log(\frac{F}{F_0})\) |