Problem 14 - Algebra for College Students

Question

Solve each exponential equation and express approximate solutions to the nearest hundredth. $$ e^{x-1}=8.2 $$

Step-by-Step Solution

Answer

The solution is approximately x = 3.10.

Step 1: Write the Equation in Logarithmic Form

The given equation is \( e^{x-1} = 8.2 \). To solve this exponential equation, you need to convert it into a logarithmic form. Since the base is \( e \), we use the natural logarithm (ln). The equation becomes \( x-1 = \ln(8.2) \).

Step 2: Solve for x

To isolate \( x \), add 1 to both sides of the equation. This results in \( x = \ln(8.2) + 1 \).

Step 3: Approximate the Solution

Now, calculate \( \ln(8.2) + 1 \) using a calculator. First, find \( \ln(8.2) \) which is approximately 2.104. Add 1 to this value to get \( x \approx 3.104 \). Round to the nearest hundredth: \( x \approx 3.10 \).

Key Concepts

Natural Logarithms

Natural logarithms provide a crucial link in solving many exponential equations. They use the mathematical constant \( e \), approximately valued at 2.71828. Whenever you encounter an equation involving \( e \), natural logarithms become particularly handy.
Natural logs are denoted by "ln". They are useful because they allow you to rewrite exponential equations in a form manageable for solving. For instance, from the original equation \( e^{x-1} = 8.2 \), you can move into logarithmic territory because \( \ln(e^y) = y \). This property helps transform and simplify exponential expressions.
Remember, converting from an exponential form \( e^y = z \) to a logarithmic form \( y = \ln(z) \) keeps equations clear and solutions easier to approximate. It empowers you to handle exponent-related questions by breaking them down into straightforward steps.

Solving Equations

Solving equations involving exponents involves a series of logical steps that transform the problem into a simpler form. With exponential equations that contain \( e \), converting to a logarithmic form is the first step. Once you've rewritten \( e^{x-1} = 8.2 \) as \( x - 1 = \ln(8.2) \), you move on to isolating the variable.
Isolation involves basic algebraic manipulation. In this case, add 1 to both sides to solve for \( x \), resulting in \( x = \ln(8.2) + 1 \). This step is critical because it prepares the equation for evaluating and approximating a numerical solution.
Each step in solving equations requires careful manipulation, remembering to maintain equality. These steps ensure you keep the variable of interest isolated, while still balancing the rest of the equation.

Logarithmic Form

Understanding how to convert equations into logarithmic form is essential for solving exponential equations effectively. Logarithmic form essentially takes an equation of the type \( b^y = z \) and converts it into \( y = \log_b(z) \). In practice, for equations with base \( e \), it becomes \( b^y = z \) to \( y = \ln(z) \).
This transform is powerful because it swaps the trouble of dealing with exponents into a simpler multiplication problem. In our specific equation, \( e^{x-1} = 8.2 \) transformed into the logarithmic form \( x - 1 = \ln(8.2) \). This format helps because logarithmic values, like \( \ln(8.2) \), can be readily calculated with a calculator, making subsequent algebraic steps easier.
Logarithms provide a pathway to simplify and exact solutions. They make solving exponent-centric equations systematically achievable, paving the path to exact numerical results without resorting to tedious trial and error.