How to simplify $\ln 2n - \ln n$ using logarithmic properties?

Conditions

• $a > 0$ and $b > 0$ (here $2n > 0$ and $n > 0$).
• The base of the logarithm is $e$ (natural logarithm).

Reasoning, step by step

1. Identify the expression as a difference of two logarithms with the same base.
2. Apply the quotient rule: $\ln 2n - \ln n = \ln \left(\frac{2n}{n}\right)$.
3. Simplify the fraction inside the logarithm: $\frac{2n}{n} = 2$.
4. Conclude that the expression equals $\ln 2$.

Example

The video states: 'we can handle these first two terms using this logarithmic property'. The screen shows $\ln 2n - \ln n$ being simplified to $\ln \frac{2n}{n}$ and then to $\ln 2$.

Common misconceptions

• Confusing the quotient rule with the product rule ($\ln a + \ln b = \ln(ab)$).
• Incorrectly simplifying $\frac{2n}{n}$ to $2n$ or $n$.

Watch the explanation

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