Solve For X X X : 3 , 125 = 5 − 10 + 3 X 3,125 = 5^{-10 + 3x} 3 , 125 = 5 − 10 + 3 X X = X = X =

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of exponential functions and their properties. In this article, we will focus on solving a specific type of exponential equation, namely the equation $3,125 = 5^{-10 + 3x}$, where we need to find the value of $x$. We will break down the solution into manageable steps, using a combination of algebraic manipulations and logarithmic properties.

Understanding Exponential Equations

Exponential equations involve exponential functions, which are functions of the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is the variable. The base $a$ can be any positive real number, but it is usually a positive integer. Exponential equations can be written in the form $a^x = b$, where $a$ and $b$ are positive real numbers.

The Given Equation

The given equation is $3,125 = 5^{-10 + 3x}$. To solve for $x$, we need to isolate the variable $x$ on one side of the equation. The first step is to rewrite the equation in a more manageable form.

Step 1: Rewrite the Equation

We can rewrite the equation as $5^{-10 + 3x} = 3,125$. The next step is to express $3,125$ as a power of $5$.

Step 2: Express $3,125$ as a Power of $5$

$3,125$ can be expressed as $5^5$. Therefore, we can rewrite the equation as $5^{-10 + 3x} = 5^5$.

Step 3: Equate the Exponents

Since the bases are the same, we can equate the exponents. This gives us the equation $-10 + 3x = 5$.

Step 4: Solve for $x$

To solve for $x$, we need to isolate the variable $x$ on one side of the equation. We can do this by adding $10$ to both sides of the equation, which gives us $3x = 15$. Then, we can divide both sides of the equation by $3$, which gives us $x = 5$.

Conclusion

In this article, we solved the exponential equation $3,125 = 5^{-10 + 3x}$ for $x$. We broke down the solution into manageable steps, using a combination of algebraic manipulations and logarithmic properties. The final answer is $x = 5$.

Additional Tips and Tricks

• When solving exponential equations, it is essential to rewrite the equation in a more manageable form.
• Expressing the constant term as a power of the base can help simplify the equation.
• Equating the exponents is a crucial step in solving exponential equations.
• Isolating the variable $x$ on one side of the equation is the final step in solving exponential equations.

Common Mistakes to Avoid

• Failing to rewrite the equation in a more manageable form can lead to confusion and errors.
• Not equating the exponents can result in incorrect solutions.
• Not isolating the variable $x$ on one side of the equation can lead to incorrect solutions.

Real-World Applications

Exponential equations have numerous real-world applications, including:

• Modeling population growth and decline
• Calculating compound interest
• Analyzing chemical reactions
• Solving problems in physics and engineering

Final Thoughts

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Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential function, which is a function of the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is the variable.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable $x$ on one side of the equation. This can be done by rewriting the equation in a more manageable form, equating the exponents, and isolating the variable $x$ on one side of the equation.

Q: What is the first step in solving an exponential equation?

A: The first step in solving an exponential equation is to rewrite the equation in a more manageable form. This can be done by expressing the constant term as a power of the base.

Q: How do I express a constant term as a power of the base?

A: To express a constant term as a power of the base, you need to find the power to which the base must be raised to equal the constant term. For example, if the constant term is $3,125$, you can express it as $5^5$.

Q: What is the next step in solving an exponential equation?

A: The next step in solving an exponential equation is to equate the exponents. This can be done by setting the exponents equal to each other and solving for the variable $x$.

Q: How do I equate the exponents?

A: To equate the exponents, you need to set the exponents equal to each other and solve for the variable $x$. For example, if the equation is $5^{-10 + 3x} = 5^5$, you can equate the exponents by setting $-10 + 3x = 5$.

Q: What is the final step in solving an exponential equation?

A: The final step in solving an exponential equation is to isolate the variable $x$ on one side of the equation. This can be done by adding or subtracting the same value to both sides of the equation, or by multiplying or dividing both sides of the equation by the same non-zero value.

Q: What are some common mistakes to avoid when solving exponential equations?

A: Some common mistakes to avoid when solving exponential equations include:

• Failing to rewrite the equation in a more manageable form
• Not equating the exponents
• Not isolating the variable $x$ on one side of the equation
• Making errors when simplifying the equation

Q: What are some real-world applications of exponential equations?

A: Exponential equations have numerous real-world applications, including:

• Modeling population growth and decline
• Calculating compound interest
• Analyzing chemical reactions
• Solving problems in physics and engineering

Q: How can I practice solving exponential equations?

A: You can practice solving exponential equations by working through examples and exercises in a textbook or online resource. You can also try solving real-world problems that involve exponential equations.

Q: What are some tips for solving exponential equations?

A: Some tips for solving exponential equations include:

• Always rewrite the equation in a more manageable form
• Equate the exponents carefully
• Isolate the variable $x$ on one side of the equation
• Check your work carefully to avoid errors

Q: What are some common exponential equations that I should know how to solve?

A: Some common exponential equations that you should know how to solve include:

• $a^x = b$
• $a^{-x} = b$
• $a^{x+y} = b$
• $a^{x-y} = b$

Q: How can I use exponential equations in real-world problems?

A: You can use exponential equations to model real-world problems such as population growth and decline, compound interest, and chemical reactions. You can also use exponential equations to solve problems in physics and engineering.