Evaluate e^(-2 natural log of 2)

Problem

$e^{-2\ln 2}$

Solution

1. Identify the expression as a base $e$ raised to a power involving a natural logarithm.

2. Apply the power rule of logarithms, which states $a\ln b=\ln b^a$ to the exponent.

$e^{-2\ln 2}=e^{\ln 2^{-2}}$

3. Simplify the exponent by calculating the value of the power.

$2^{-2}=\frac{1}{2^2}=\frac{1}{4}$

4. Substitute the simplified value back into the expression.

$e^{\ln (1/4)}$

5. Apply the inverse property of logarithms and exponentials, $e^{\ln x}=x$ to find the final value.

$e^{\ln (1/4)}=\frac{1}{4}$

Final Answer

$e^{-2\ln 2}=\frac{1}{4}$

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