Question

Apply the inverse properties of \(\ln x\) and \(e^{x}\) to simplify the given expression. $$ e^{\ln \sqrt{x}} $$

Short answer

The simplified form of the expression \(e^{\ln \sqrt{x}}\) is \(x^{1/2}\), or the square root of \(x\)

Step-by-step solution

Identify the properties

To begin simplifying the expression, identify the inverse properties of natural logarithms and exponentials, and the property of square roots that square root of x equals x to the 1/2 power.

Simplify the Square Root Expression

Next, rewrite the square root of x as \(x^{1/2}\) in the expression. Therefore, it becomes: \(e^{\ln x^{1/2}}\).

Apply the Inverse Properties

Then, apply the inverse property of \(e^{x}\) and \(\ln x\) which states that: \(e^{\ln x} = x\). Using this property with \(x^{1/2}\), we get: \(e^{\ln x^{1/2}} = x^{1/2}\)