Solve The Equation For $x$.$e^{12x + 12} = E^{-6x}$Show Your Work Below.
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Introduction
Exponential equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving the equation $e^{12x + 12} = e^{-6x}$. We will break down the solution into manageable steps, making it easy to follow along.
Step 1: Set Up the Equation
The given equation is $e^{12x + 12} = e^{-6x}$. To solve for $x$, we need to isolate the variable. The first step is to set up the equation by equating the exponents.
Step 2: Equate the Exponents
Since the bases of the exponential expressions are the same (both are $e$), we can equate the exponents. This gives us the equation $12x + 12 = -6x$.
Step 3: Simplify the Equation
Now that we have the equation $12x + 12 = -6x$, we can simplify it by combining like terms. We can add $6x$ to both sides of the equation to get $18x + 12 = 0$.
Step 4: Isolate the Variable
To isolate the variable $x$, we need to get rid of the constant term on the left-hand side of the equation. We can do this by subtracting $12$ from both sides of the equation, giving us $18x = -12$.
Step 5: Solve for $x$
Now that we have the equation $18x = -12$, we can solve for $x$ by dividing both sides of the equation by $18$. This gives us $x = -\frac{12}{18}$.
Step 6: Simplify the Solution
We can simplify the solution $x = -\frac{12}{18}$ by dividing both the numerator and the denominator by their greatest common divisor, which is $6$. This gives us $x = -\frac{2}{3}$.
Conclusion
In this article, we solved the exponential equation $e^{12x + 12} = e^{-6x}$ by setting up the equation, equating the exponents, simplifying the equation, isolating the variable, and solving for $x$. The final solution is $x = -\frac{2}{3}$.
Tips and Tricks
When solving exponential equations, it's essential to remember that the bases of the exponential expressions must be the same. If the bases are different, you cannot equate the exponents. Additionally, when simplifying the equation, be sure to combine like terms and isolate the variable to make it easier to solve for the unknown.
Real-World Applications
Exponential equations have numerous real-world applications, including population growth, chemical reactions, and financial modeling. In these contexts, solving exponential equations can help us understand and predict the behavior of complex systems.
Common Mistakes
When solving exponential equations, some common mistakes to avoid include:
- Not equating the exponents when the bases are the same
- Not simplifying the equation by combining like terms
- Not isolating the variable to make it easier to solve for the unknown
Final Thoughts
Solving exponential equations requires a step-by-step approach, but with practice and patience, it can become second nature. By following the steps outlined in this article, you can confidently solve exponential equations and apply them to real-world problems.
Additional Resources
For more information on solving exponential equations, check out the following resources:
- Khan Academy: Exponential Equations
- Mathway: Exponential Equations Solver
- Wolfram Alpha: Exponential Equations Calculator
Frequently Asked Questions
Q: What is the difference between an exponential equation and a linear equation? A: An exponential equation involves a variable in the exponent, while a linear equation involves a variable in the base.
Q: How do I know when to use the product rule and when to use the quotient rule when solving exponential equations? A: Use the product rule when multiplying exponential expressions with the same base, and use the quotient rule when dividing exponential expressions with the same base.
Q: Can I use the same steps to solve logarithmic equations as I would to solve exponential equations?
A: No, logarithmic equations require a different approach than exponential equations.
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Introduction
Exponential equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will answer some of the most frequently asked questions about exponential equations, providing you with a deeper understanding of these complex equations.
Q: What is an exponential equation?
A: An exponential equation is an equation that involves a variable in the exponent. It is typically 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 know when to use the product rule and when to use the quotient rule when solving exponential equations?
A: Use the product rule when multiplying exponential expressions with the same base, and use the quotient rule when dividing exponential expressions with the same base.
Q: Can I use the same steps to solve logarithmic equations as I would to solve exponential equations?
A: No, logarithmic equations require a different approach than exponential equations. Logarithmic equations involve the inverse operation of exponentiation, and require a different set of rules and techniques to solve.
Q: What is the difference between an exponential equation and a linear equation?
A: An exponential equation involves a variable in the exponent, while a linear equation involves a variable in the base.
Q: How do I solve an exponential equation with a negative exponent?
A: To solve an exponential equation with a negative exponent, you can rewrite the equation with a positive exponent by taking the reciprocal of both sides.
Q: Can I use a calculator to solve exponential equations?
A: Yes, you can use a calculator to solve exponential equations. However, it's essential to understand the underlying math and be able to verify the solution using algebraic methods.
Q: What is the significance of the base in an exponential equation?
A: The base in an exponential equation determines the rate of growth or decay of the function. A base greater than 1 represents exponential growth, while a base between 0 and 1 represents exponential decay.
Q: Can I use exponential equations to model real-world phenomena?
A: Yes, exponential equations can be used to model a wide range of real-world phenomena, including population growth, chemical reactions, and financial modeling.
Q: How do I determine the domain of an exponential function?
A: The domain of an exponential function is all real numbers, unless the base is negative, in which case the domain is restricted to non-negative real numbers.
Q: Can I use exponential equations to solve problems involving finance?
A: Yes, exponential equations can be used to solve problems involving finance, such as calculating compound interest or determining the future value of an investment.
Q: What is the difference between an exponential function and a power function?
A: An exponential function involves a variable in the exponent, while a power function involves a variable in the base.
Q: Can I use exponential equations to solve problems involving science and engineering?
A: Yes, exponential equations can be used to solve problems involving science and engineering, such as modeling population growth or determining the half-life of a radioactive substance.
Q: How do I determine the range of an exponential function?
A: The range of an exponential function is all positive real numbers, unless the base is negative, in which case the range is restricted to non-positive real numbers.
Q: Can I use exponential equations to solve problems involving economics?
A: Yes, exponential equations can be used to solve problems involving economics, such as modeling economic growth or determining the impact of inflation on a country's economy.
Conclusion
Exponential equations are a powerful tool for modeling and solving a wide range of problems in mathematics, science, and engineering. By understanding the underlying math and being able to apply the correct techniques, you can confidently solve exponential equations and apply them to real-world problems.
Additional Resources
For more information on exponential equations, check out the following resources:
- Khan Academy: Exponential Equations
- Mathway: Exponential Equations Solver
- Wolfram Alpha: Exponential Equations Calculator
Frequently Asked Questions
Q: What is the difference between an exponential equation and a linear equation? A: An exponential equation involves a variable in the exponent, while a linear equation involves a variable in the base.
Q: How do I know when to use the product rule and when to use the quotient rule when solving exponential equations? A: Use the product rule when multiplying exponential expressions with the same base, and use the quotient rule when dividing exponential expressions with the same base.
Q: Can I use the same steps to solve logarithmic equations as I would to solve exponential equations? A: No, logarithmic equations require a different approach than exponential equations.