Problem 10 - Precalculus with Limits

Question

Solve for \(x.\) \(e^{x}=2\)

Step-by-Step Solution

Answer

\(\ln{(2)}\)

Step 1: Write down the equation

The equation given is \(e^{x}=2\).

Step 2: Take the natural logarithm (ln) of both sides

The natural logarithm is the inverse of the exponential function (e), so it provides a way for us to decrease the power and bring \(x\) down. This is done using the log rule: ln(\(e^{x}\))=\(x\). We apply the natural logarithm (ln) on both sides of the equation, converting the equation to \(\ln{(e^{x})} = \ln{2}\).

Step 3: Simplify equation

Using the rule mentioned in the previous step, ln(\(e^{x}\)) simplifies to \(x\). Thus the equation becomes \(x = \ln{2}\).

Key Concepts

Natural Logarithms

Natural logarithms, represented as \(\ln\), are logarithms with the base \(e\), where \(e\) is an important mathematical constant approximately equal to 2.718. They are incredibly useful in simplifying exponential equations. For instance, in the given equation \(e^{x} = 2\), taking the natural logarithm of both sides helps to solve for \(x\). This occurs because the natural logarithm and the exponential function \(e\) are inverse functions. Thus, when you take \(\ln(e^{x})\), it simplifies to \(x\).

• The natural logarithm is primarily used when dealing with exponential growth or decay problems.
• It is especially prevalent in fields such as finance, physics, and most of the natural sciences.

Applying natural logarithms can transform complex equations into easier, solvable forms. This aspect makes them a favorite tool among mathematicians and scientists alike.

Inverse Functions

An inverse function essentially "undoes" the action of the original function, meaning if a function is applied and then its inverse is applied, you'll end up back where you started. For example, with the exponential function \(e^{x}\), its inverse is the natural logarithm, \(\ln(x)\). This relationship is crucial in solving equations like \(e^{x} = 2\) since applying \(\ln\) reverses the effect of exponentiation.

• When you have \(\ln(e^{x})\), it simplifies directly to \(x\).
• Understanding inverse functions can greatly simplify solving equations and provide insights into the properties of functions themselves.

Inverse functions have applications in many areas of mathematics and science, including finding angles in trigonometry or reversing signals in engineering. Mastering their use can greatly enhance one's mathematical toolkit.

Solving Equations with Logarithms

Solving equations using logarithms is a common method when dealing with exponential equations. In the context of the equation \(e^{x} = 2\), the approach involves applying the logarithm to both sides. Here, we use the natural logarithm because it is the inverse of the exponential base \(e\).

• This method turns the exponentiated variable into a simple linear form \(x = \ln(2)\).
• Always ensure the logarithm used matches the base of the exponential part for straightforward solutions.

This approach is powerful as it transforms a seemingly complex problem into something as simple as computing the logarithm of a number. It highlights the functionality and versatility of logarithms in mathematics. For those working with exponential growth or decay, like in population models or radioactive decay, logarithms provide a valuable technique to derive solutions directly.