Problem 70 - College Algebra with Modeling and Visualization

Question

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. $$1-2 e^{x}=-5$$

Step-by-Step Solution

Answer

The solution for \(x\) is approximately 0.69.

Step 1: Isolate the Exponential Expression

Start by isolating the exponential part of the equation. The equation is \(1 - 2e^x = -5\). Add 1 to both sides to get \(-2e^x = -4\). Divide both sides by -2 to isolate \(e^x\), resulting in \(e^x = 2\).

Step 2: Apply the Natural Logarithm

To solve for \(x\), take the natural logarithm (ln) of both sides of the equation. This gives us \(\ln(e^x) = \ln(2)\). Since \(\ln(e^x) = x\), we can simplify this to \(x = \ln(2)\).

Step 3: Approximate the Natural Logarithm

Use a calculator to approximate \(\ln(2)\) to the nearest hundredth. The value of \(\ln(2)\) is approximately 0.693. Therefore, \(x \approx 0.69\).

Key Concepts

Natural Logarithm

Understanding the natural logarithm is a key step in solving exponential equations. The natural logarithm, usually written as \( \ln \), is a type of logarithm that uses \( e \) (approximately 2.718) as the base. This mathematical constant \( e \) is important because it appears in many natural processes, such as compound interest and population growth.
When you have an equation involving the natural exponential function \( e^x \), you can use the natural logarithm to solve for \( x \). For instance, if you have \( e^x = a \), taking the natural logarithm of both sides will yield \( \ln(e^x) = \ln(a) \), which simplifies to \( x = \ln(a) \) due to the property \( \ln(e^x) = x \).
This process transforms a potentially complex equation into a simple linear equation, making it a powerful tool for solving exponential equations.

Change of Base Formula

The change of base formula is essential when you need to work with logarithms that aren't in a convenient base for your specific problem. In the context of this exercise, the natural logarithm basis \( e \) is already suitable given the presence of the exponential function \( e^x \).
However, if you needed to convert this logarithmic expression into another base, such as base 10, the change of base formula becomes handy. It states that:

• For any logarithmic expression \( \log_b(a) \), you can convert it to a new base \( c \) using the formula: \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \).

This means you can calculate log values in any base using a more readily available base from your calculator or settings. This flexibility simplifies calculations for different logarithms and helps solve equations when different operations or bases are involved.

Solving Equations

Solving exponential equations often involves isolating the exponential part and then using logarithms to cancel out the exponential function. Let's break it down with an example:
Consider the equation \( 1 - 2e^x = -5 \). The first step is to isolate \( e^x \). Add 1 to both sides of the equation to get \( -2e^x = -4 \) and then divide by -2, resulting in \( e^x = 2 \).
With \( e^x \) isolated, apply the natural logarithm to both sides: \( \ln(e^x) = \ln(2) \). The property \( \ln(e^x) = x \) allows you to simplify to \( x = \ln(2) \).
The result \( x = \ln(2) \) can be easily approximated using a calculator, giving \( x \approx 0.69 \), which is rounded to the nearest hundredth. This process demonstrates the power of natural logarithms in simplifying and solving exponential equations efficiently.