Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. ln(e²/5)

Verified step by step guidance

1. Recall the logarithmic identity that relates the natural logarithm and exponentials: \(\ln\left(e^x\right) = x\). This means the natural logarithm of \(e\) raised to any power simplifies directly to that power.
2. Identify the exponent in the given expression \(\ln\left(e^{2/5}\right)\). Here, the exponent is \(\frac{2}{5}\).
3. Apply the identity by rewriting the expression as \(\ln\left(e^{2/5}\right) = \frac{2}{5}\).
4. Since the expression simplifies directly to a number, no further expansion is needed.
5. Thus, the expanded form of \(\ln\left(e^{2/5}\right)\) is simply the exponent \(\frac{2}{5}\).

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules that simplify logarithmic expressions. For example, the power rule states that ln(a^b) = b * ln(a), which is essential for expanding and simplifying expressions like ln(e^(2/5)).

Natural Logarithm and the Number e

The natural logarithm, denoted ln, is the logarithm with base e, where e is approximately 2.718. A key property is that ln(e^x) = x, which allows direct simplification of expressions involving e raised to a power inside a logarithm.

Evaluating Logarithmic Expressions Without a Calculator

Some logarithmic expressions can be simplified exactly using known properties and values, avoiding the need for a calculator. Recognizing forms like ln(e^x) helps evaluate expressions quickly and accurately by reducing them to their exponents.