Solving Exponential Equations: A Step-by-Step Guide

by Alex Johnson 52 views

Hey there, math enthusiasts! Today, we're diving into the exciting world of exponential equations. These equations pop up everywhere, from modeling population growth to understanding radioactive decay. Let's tackle a specific problem: solving the exponential equation 5e^(5x) = 1945 and then finding its decimal approximation using a calculator. This guide will break down the process step-by-step, making it easy for you to understand and apply these concepts.

Unveiling Exponential Equations

First things first, what exactly is an exponential equation? It's an equation where the variable appears in the exponent. This means the variable is part of the power to which a base number is raised. In our example, 5e^(5x) = 1945, x is in the exponent of e. The constant e is Euler's number, approximately equal to 2.71828, a crucial constant in mathematics, especially in calculus and exponential growth/decay models. Understanding this is key to solving these types of equations. You will encounter exponential equations when dealing with various real-world scenarios, such as compound interest, population dynamics, and radioactive decay. The ability to manipulate and solve these equations is a valuable skill in many fields. Let us understand the concept of the exponential equation with its essential components. Exponential equations have a base, an exponent, and a coefficient. The base is the number that is being raised to a power (exponent). In our case, the base is e, which is Euler's number. The exponent is the power to which the base is raised. Here, the exponent is 5x. The coefficient is the number that multiplies the exponential term, it is 5 in this equation. The goal when solving an exponential equation is to isolate the exponential term and then use logarithms to solve for the variable.

Core Concepts

  • Base: The number being raised to a power (e.g., in e^(5x), the base is e).
  • Exponent: The power to which the base is raised (e.g., 5x in e^(5x)).
  • Coefficient: The number multiplying the exponential term (e.g., 5 in 5e^(5x)).
  • Euler's Number (e): A mathematical constant, approximately 2.71828, essential in calculus and exponential functions.

Step-by-Step Solution

Now, let's solve the equation 5e^(5x) = 1945 step-by-step. Remember, the goal is to isolate the variable x. Let's break it down into manageable parts. Solving an exponential equation often involves several steps to isolate the variable. The first step typically involves isolating the exponential term.

Step 1: Isolate the Exponential Term

Our first step is to isolate the exponential term, which is e^(5x). To do this, we need to get rid of the coefficient 5. Divide both sides of the equation by 5:

5e^(5x) / 5 = 1945 / 5

This simplifies to:

e^(5x) = 389

We have successfully isolated the exponential term! This step is fundamental, as it sets the stage for using logarithms. Remember, the key is to perform the same operation on both sides of the equation to maintain balance.

Step 2: Apply the Natural Logarithm

Now that we have e^(5x) = 389, we need to get the exponent 5x down from the power. We'll use the natural logarithm (ln), which is the inverse of the exponential function with base e. Take the natural logarithm of both sides:

ln(e^(5x)) = ln(389)

Using the property of logarithms that ln(e^a) = a, the left side simplifies to:

5x = ln(389)

This step is crucial because it allows us to 'bring down' the exponent and solve for x. Applying the natural logarithm effectively removes the exponential part, allowing us to solve for the exponent. The natural logarithm is a powerful tool in solving exponential equations.

Step 3: Solve for x

We're almost there! Now, we have the equation 5x = ln(389). To isolate x, divide both sides by 5:

x = ln(389) / 5

This is the exact solution for x. It's expressed in terms of the natural logarithm. It’s important to understand the concept of isolating the variable x to obtain a solution.

Step 4: Calculate the Decimal Approximation

To find the decimal approximation, use a calculator to evaluate ln(389) and then divide by 5. Make sure your calculator is in the natural logarithm mode (usually labeled as