Problem 3 - College Algebra

Question

Solve each exponential equation in Exercises \(1-26\) Express the solution set in terms of natural logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$e^{x}=5.7$$

Step-by-Step Solution

Answer

After applying the natural logarithm to both sides, the solution to the exponential equation \(e^{x} = 5.7\) is \(x = ln(5.7)\). The decimal approximation of the solution, rounded to two decimal places, will be found using a calculator in step 2.

Step 1: Apply Natural Logarithm to Both Sides

The goal here is to isolate the x. Therefore, we'll apply the natural logarithm function (ln) to both sides of the equation, which will leave us with \(x = ln(5.7)\). This is possible because the natural logarithm of e raised to the x, \(ln(e^{x})\), is simply x.

Step 2: Find the Numerical Approximation

Use a calculator to approximate the value of \(ln(5.7)\). The calculator should be set to round to two decimal places

Key Concepts

Natural Logarithms

Natural logarithms can be thought of as a way to undo the action of an exponential function with base e, where e is the irrational number roughly equal to 2.71828. When you see an equation like \(e^{x} = 5.7\), the exponential part (\(e^{x}\)) can be 'unwrapped' by taking the natural logarithm of both sides. Why? Because the natural logarithm and the exponential function with base e are inverse operations. This means that \(\ln(e^{x}) = x\). It's like saying 'what power do we need to raise e to, to get e to the x?' The answer is clearly x.

Understanding this concept is fundamental when solving exponential equations, as it helps to 'bring down' the variable from the exponent. In practice, to use natural logarithms to solve for x, you would apply \(\ln\) to both sides of the equation. This is a powerful algebraic tool that simplifies otherwise complex logarithmic and exponential relationships.

Numerical Approximation

Numerical approximation comes into play when exact answers are not manageable or not necessary. This can be the case with natural logarithms because we're often dealing with irrational numbers that cannot be expressed precisely with a finite number of decimal places. After isolating the variable, as we did in the previous step with \(x = \ln(5.7)\), we're left with a natural log function that usually won't spit out a nice, clean integer.

To solve this in real-world applications, you would use a scientific calculator or software that can give you a decimal approximation of the value. Bear in mind that approximations are just that – not exact, but close enough for practical purposes. They're usually rounded to a certain number of significant figures or decimal places, and in this case, we want the value correct to two decimal places.

Isolate the Variable

To solve an equation, you need to 'isolate' the variable, which means manipulating the equation to get the variable alone on one side. With simple equations, this might involve basic operations like addition or subtraction. But with exponential equations, the process involves using logarithms.

Isolating the variable makes an equation simpler and the solution clear. In our example \(e^{x} = 5.7\), by applying a natural logarithm to both sides, we isolated x on one side, giving us \(x = \ln(5.7)\). This is important because it turns the problem from a transcendental to an algebraic equation, something that is more straightforward to solve. Remember, the goal is to perform the same operation on both sides of the equation to maintain its equality while rearranging it so that the variable stands alone.