Common Logarithm 856C4F

1. The problem is to understand the difference between the common logarithm and the natural logarithm. 2. The common logarithm is the logarithm with base 10, denoted as $\log_{10}(x)$ or simply $\log(x)$ in many contexts. 3. The natural logarithm is the logarithm with base $e$ (where $e \approx 2.71828$), denoted as $\ln(x)$. 4. Important rule: $\log_{10}(x)$ and $\ln(x)$ are different functions because they have different bases. 5. For example, $\log_{10}(100) = 2$ because $10^2 = 100$, but $\ln(100) \approx 4.605$ because $e^{4.605} \approx 100$. 6. To convert between them, use the formula: $$\log_{10}(x) = \frac{\ln(x)}{\ln(10)}$$ or equivalently, $$\ln(x) = \log_{10}(x) \times \ln(10)$$ 7. This means you can calculate one if you know the other and the value of $\ln(10)$. 8. Summary: "common log" means logarithm base 10, "natural log" means logarithm base $e$. Final answer: Common log is $\log_{10}(x)$, natural log is $\ln(x)$, and they differ by their bases.