Simplify e^(1/2 natural log of x)

Solution:

Step 1:

Rewrite the expression $\frac{1}{2} \ln(x)$ by applying the power rule of logarithms to get $e^{\ln(x^{\frac{1}{2}})}$.

Step 2:

Recognize that raising $e$ to the power of a natural log cancels out, leaving us with $x^{\frac{1}{2}}$.

Knowledge Notes:

The problem at hand involves simplifying an expression that contains an exponentiated natural logarithm. The key knowledge points to understand this problem are:

1. Logarithm Properties: Logarithms have several properties that simplify expressions. One such property is the power rule, which states that $a \ln(b) = \ln(b^a)$. This allows us to move constants inside the logarithm as an exponent of its argument.

2. Inverse Functions: The natural logarithm function, denoted as $\ln(x)$, is the inverse of the exponential function with base $e$, written as $e^x$. This means that $\ln(e^x) = x$ and $e^{\ln(x)} = x$. When an exponentiation by $e$ is followed by a natural logarithm (or vice versa), they cancel each other out.

3. Exponents and Radicals: The expression $x^{\frac{1}{2}}$ is equivalent to the square root of $x$, denoted as $\sqrt{x}$. This is because fractional exponents represent roots, with the denominator indicating the type of root (in this case, a square root).

4. Simplifying Expressions: The process of simplifying expressions often involves recognizing and applying these properties to transform complex expressions into simpler or more familiar forms. In this case, the simplification process reduces an exponentiated logarithm to a radical expression.

By understanding these concepts, one can approach and solve problems involving logarithms and exponents systematically and efficiently.