Question

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(e^{2 x}=50\)

Short answer

The solution to the equation \(e^{2x} = 50\) is \(x = \ln(50) / 2\). After approximating, it is \(x = 1.918\).

Step-by-step solution

Take Natural Logarithm (ln) on Both Sides

By applying logarithmic rules, one can take that natural logarithm of both sides of the equation without changing the equality. When one takes the natural log of \(e^{2x}\), the logarithm simplifies in such a way that the variable leaves the exponent. This will allow us to isolate \(x\). This results in: \(\ln( e^{2x} ) = \ln(50)\).

Simplify the Left Side

According to logarithmic rules, \( \ln(e^{2x}) \) simplifies to \( 2x \). This simplification occurs because the ln and the \(e\) cancel out. Now we have \(2x = \ln(50)\).

Solve for x

To solve for \(x\), divide both sides of the equation by 2. This gives us \(x = \ln(50) / 2\).

Approximate the Value

Using a calculator that can compute ln, input \(\ln(50)/2\). Make sure to round off to the nearest three decimal places.

Key concepts

Natural Logarithm

The natural logarithm, often denoted as "ln," is a logarithm with the base of the mathematical constant, Euler's number \(e\), which is approximately 2.71828. In calculus and higher mathematics, the natural logarithm is widely used because it simplifies many mathematical expressions.
The key function of the natural logarithm is to "undo" exponentials. For example, taking the natural log of \(e^x\) results in simply \(x\). This property is what makes natural logarithms very powerful. They're the go-to choice in solving exponential equations.
Understanding natural logarithms is crucial for algebraically solving equations that involve the exponential function like \(e^{2x}=50\). By applying the natural log, you can transform an equation into a more workable form.

Logarithmic Rules

Logarithmic rules are essential when working with exponential equations. In the original exercise, the logarithmic identity \( \ln(e^a) = a \) is used. This specific identity helps to "bring down" the exponent when taking the natural log of an exponential expression.
Some other important logarithmic rules include:

• \( \ln(xy) = \ln(x) + \ln(y) \) - Helps in breaking down complex expressions.
• \( \ln(x/y) = \ln(x) - \ln(y) \) - Assists in simplifying division within logs.
• \( \ln(x^n) = n \ln(x) \) - Allows reduction of powers inside the log.

Using these rules efficiently allows for simplifying equations and isolating variables, which leads to solving the equation accurately.

Algebraic Isolation

Algebraic isolation involves solving for a variable by itself on one side of the equation. In other words, you want to "isolate" the variable of interest. In the given equation \(e^{2x}=50\), our goal is to solve for \(x\).
After applying the natural logarithm to both sides, you simplify using the rule \(\ln(e^{2x}) = 2x\). This step is crucial because it reduces the equation to \(2x = \ln(50)\).
Isolating \(x\) involves simple algebraic manipulation: divide both sides by 2 yielding \(x = \ln(50)/2\). This process of isolating the variable makes complex equations manageable and easier to solve.

Approximation Techniques

In mathematics, approximation techniques are used for estimating values to a desired level of accuracy, especially when exact computation is impractical.
With our equation \(x = \ln(50)/2\), you would use a calculator to find the natural logarithm of 50, then divide the result by 2.
Keep in mind that you are required to round this result to three decimal places. Rounding helps display the number in a more compact form, making it easier to interpret.

• Use a calculator or software that can handle natural logarithms.
• Perform the computation as accurately as possible and write down the result rounded to the desired number of decimal places.
• Double-check your rounding for accuracy.

This process helps deal with irrational numbers and ensures that your solution is precise enough for practical purposes.