Find each value. If applicable, give an approximation to four decimal places. ln 1/e²

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Recall the properties of logarithms, especially the natural logarithm (ln). The natural logarithm of a quotient can be written as the difference of logarithms: \(\ln \left( \frac{a}{b} \right) = \ln a - \ln b\).

Rewrite the expression \(\ln \left( \frac{1}{e^2} \right)\) using the logarithm property: \(\ln 1 - \ln e^2\).

Evaluate \(\ln 1\). Since the natural logarithm of 1 is always 0, we have \(\ln 1 = 0\).

Evaluate \(\ln e^2\). Using the property \(\ln e^x = x\), this simplifies to \(2\).

Combine the results: \(0 - 2 = -2\). This is the exact value of the expression.

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Natural Logarithm (ln)

The natural logarithm, denoted as ln, is the logarithm to the base e, where e ≈ 2.718. It answers the question: to what power must e be raised to get a given number? For example, ln(e) = 1 because e^1 = e.

Properties of Exponents

Exponents represent repeated multiplication. Key properties include e^a * e^b = e^(a+b) and (e^a)^b = e^(ab). Understanding these helps simplify expressions inside logarithms, such as rewriting 1/e^2 as e^(-2).

Logarithm of a Power

The logarithm of a power follows the rule ln(a^b) = b * ln(a). This allows simplification of logarithmic expressions by bringing the exponent down as a multiplier, making calculations easier, especially when the base matches the logarithm's base.

Find each value. If applicable, give an approximation to four decimal places. ln 1/e2