Question

In Exercises, solve the equation for \(x\). $$ e^{\sqrt{x}}=e^{3} $$

Short answer

The solution to the equation \(e^{\sqrt{x}}=e^{3}\) is \(x=9\).

Step-by-step solution

Set the exponents equal to each other

Given that the bases are the same, \(e^{\sqrt{x}}=e^{3}\), it can be deduced that their exponents must be equal. So, it can be rewritten as \(\sqrt{x}=3\).

Square both sides of the equation

To get rid of the square root on the left side, square both sides of the equation. Doing so, the equation becomes \(x=9\). Squaring gets rid of the square root, but remember that squaring can introduce extraneous solutions. This arises when we 'square off' a quantity that was actually negative - which we can't have under a square root sign.

Check for extraneous solutions

Substitute \(x=9\) into the original equation to make sure it is not an extraneous root. Substituting \(x=9\) in the original equation \(e^{\sqrt{x}}=e^{3}\), we get \(e^{\sqrt{9}}=e^{3}\) that simplifies to \(e^{3}=e^{3}\). Hence, \(x=9\) is a valid solution.

Key concepts

Understanding Exponents

Exponents are a way to represent repeated multiplication of the same number. In an expression such as \(e^{3}\), \(e\) is the base, and \(3\) is the exponent, which tells us to multiply \(e\) by itself three times. This concept is fundamental when tackling exponential equations because if two expressions with the same base are equal, their exponents must be equal too.

• The notation \(a^b\) signifies that \(a\) is multiplied by itself \(b\) times.
• An equation with an exponential term, like \(e^{\sqrt{x}}= e^3\), requires setting the exponents equal to solve it, because the bases are the same.

Understanding how exponents work helps solve more complex equations efficiently. It simplifies equations to a form that is much easier to deal with, allowing you to focus on the real number calculations involved.

Identifying Extraneous Solutions

Extraneous solutions are results that emerge from the process of manipulating an equation but do not satisfy the original equation. They often arise when you perform operations like squaring both sides, which can introduce new solutions that weren’t possible in the original context.

• For instance, to remove a square root, you square both sides of the equation. However, this process can yield solutions that don't fit the original problem constraints.
• This is why it's crucial to substitute solutions back into the original equation, like checking if \(x=9\) satisfies \(e^{\sqrt{x}}=e^{3}\). If it does, then it is indeed a valid solution.

When solving equations, identifying and rejecting extraneous solutions ensures the solutions are accurate and consistent with the original equation.

Working with Square Roots

Square roots are used to determine what number, when multiplied by itself, yields the original number. In our problem, \(\sqrt{x}\) asks which number squared would result in \(x\).

• The square root of a number is often denoted by the radical symbol \(\sqrt{}\). For example, \(\sqrt{9} = 3\), because \(3^2 = 9\).
• In the equation \(\sqrt{x} = 3\), squaring both sides helps eliminate the square root, turning the equation into \(x = 9\).

Understanding square roots is critical when solving equations involving them since they can indicate multiple potential solutions, often termed the principal square root (positive) and sometimes the negative root in broader contexts. However, in this problem, since we're dealing with \(e^\) functions which traditionally consider the principal root, \(x=9\) suffices as a valid solution.