Solve The Equation For { X $} : : : { 9 = -5 + 2 E^{x / 5} \}
Introduction
Exponential equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving the equation 9 = -5 + 2e^(x/5), where e is the base of the natural logarithm. We will break down the solution into manageable steps, using algebraic manipulations and logarithmic properties to isolate the variable x.
Understanding the Equation
The given equation is 9 = -5 + 2e^(x/5). To solve for x, we need to isolate the exponential term e^(x/5). The first step is to simplify the equation by combining like terms.
Step 1: Simplify the Equation
We can start by adding 5 to both sides of the equation:
9 + 5 = 2e^(x/5)
This simplifies to:
14 = 2e^(x/5)
Step 2: Isolate the Exponential Term
Next, we need to isolate the exponential term e^(x/5). We can do this by dividing both sides of the equation by 2:
14/2 = e^(x/5)
This simplifies to:
7 = e^(x/5)
Step 3: Take the Natural Logarithm
Now that we have isolated the exponential term, we can take the natural logarithm (ln) of both sides of the equation:
ln(7) = ln(e^(x/5))
Using the property of logarithms that states ln(e^x) = x, we can simplify the right-hand side of the equation:
ln(7) = x/5
Step 4: Solve for x
Finally, we can solve for x by multiplying both sides of the equation by 5:
5 * ln(7) = x
This gives us the solution:
x = 5 * ln(7)
Conclusion
In this article, we have solved the equation 9 = -5 + 2e^(x/5) using algebraic manipulations and logarithmic properties. We have broken down the solution into manageable steps, isolating the exponential term e^(x/5) and then taking the natural logarithm to solve for x. The final solution is x = 5 * ln(7).
Example Use Cases
Exponential equations like the one we solved in this article have many practical applications in fields such as physics, engineering, and economics. For example:
- In physics, exponential equations are used to model population growth and decay, radioactive decay, and chemical reactions.
- In engineering, exponential equations are used to model the behavior of electrical circuits, mechanical systems, and thermal systems.
- In economics, exponential equations are used to model the growth of economies, the behavior of financial markets, and the spread of diseases.
Tips and Tricks
When solving exponential equations, it's essential to remember the following tips and tricks:
- Always start by simplifying the equation and isolating the exponential term.
- Use logarithmic properties to simplify the equation and solve for the variable.
- Be careful when taking the natural logarithm of both sides of the equation, as this can introduce extraneous solutions.
- Always check your solution by plugging it back into the original equation.
Conclusion
Q: What is an exponential equation?
A: An exponential equation is an equation that contains an exponential term, which is a term that involves a base raised to a power. Exponential equations can be written in the form a^x = b, where a is the base, x is the exponent, and b is the result.
Q: How do I solve an exponential equation?
A: To solve an exponential equation, you need to isolate the exponential term and then use logarithmic properties to solve for the variable. The steps involved in solving an exponential equation are:
- Simplify the equation and isolate the exponential term.
- Take the natural logarithm of both sides of the equation.
- Use the property of logarithms that states ln(e^x) = x to simplify the equation.
- Solve for the variable.
Q: What is the difference between an exponential equation and a logarithmic equation?
A: An exponential equation is an equation that contains an exponential term, while a logarithmic equation is an equation that contains a logarithmic term. Exponential equations are used to model growth and decay, while logarithmic equations are used to model the inverse of exponential growth and decay.
Q: How do I know when to use an exponential equation versus a logarithmic equation?
A: You should use an exponential equation when you are modeling growth or decay, and you should use a logarithmic equation when you are modeling the inverse of exponential growth or decay.
Q: What is the base of the natural logarithm?
A: The base of the natural logarithm is e, which is approximately equal to 2.71828.
Q: How do I use the natural logarithm to solve an exponential equation?
A: To use the natural logarithm to solve an exponential equation, you need to take the natural logarithm of both sides of the equation. This will allow you to use the property of logarithms that states ln(e^x) = x to simplify the equation.
Q: What is the difference between the natural logarithm and the common logarithm?
A: The natural logarithm is the logarithm to the base e, while the common logarithm is the logarithm to the base 10. The natural logarithm is used more frequently in mathematics and science, while the common logarithm is used more frequently in engineering and economics.
Q: How do I solve an exponential equation with a base other than e?
A: To solve an exponential equation with a base other than e, you need to use the change of base formula, which states that log_a(b) = ln(b) / ln(a). This will allow you to convert the equation to a form that can be solved using the natural logarithm.
Q: What are some common applications of exponential equations?
A: Exponential equations have many practical applications in fields such as physics, engineering, and economics. Some common applications include:
- Modeling population growth and decay
- Modeling radioactive decay
- Modeling chemical reactions
- Modeling the behavior of electrical circuits
- Modeling the growth of economies
Q: What are some common mistakes to avoid when solving exponential equations?
A: Some common mistakes to avoid when solving exponential equations include:
- Not simplifying the equation before taking the natural logarithm
- Not using the property of logarithms that states ln(e^x) = x
- Not checking the solution by plugging it back into the original equation
- Not using the change of base formula when the base is not e.