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

$$$$

Introduction

Mathematics is a vast and fascinating subject that encompasses various branches, including algebra, geometry, and calculus. One of the fundamental aspects of mathematics is solving equations, which involves finding the value of unknown variables that satisfy a given equation. In this article, we will focus on solving an exponential equation involving a base of 5, which is a common base used in mathematics.

Understanding the Equation

The given equation is $3,125 = 5^{-10 + 3x}$. To solve this equation, we need to isolate the variable x. The first step is to rewrite 3,125 as a power of 5. We know that $5^5 = 3125$, so we can rewrite the equation as $5^5 = 5^{-10 + 3x}$.

Using Exponent Properties

Now that we have rewritten the equation, we can use exponent properties to simplify it. When two powers with the same base are equal, their exponents must be equal. Therefore, we can set the exponents equal to each other: $5 = -10 + 3x$.

Solving for x

Now that we have a linear equation, we can solve for x. To do this, we need to isolate x on one side of the equation. We can start by adding 10 to both sides of the equation: $5 + 10 = 3x$. This simplifies to $15 = 3x$.

Isolating x

Now that we have $15 = 3x$, we can isolate x by dividing both sides of the equation by 3: $x = \frac{15}{3}$.

Simplifying the Expression

The expression $\frac{15}{3}$ can be simplified by dividing the numerator and denominator by their greatest common divisor, which is 3. Therefore, $x = \frac{15}{3} = 5$.

Conclusion

In this article, we solved the exponential equation $3,125 = 5^{-10 + 3x}$ for x. We started by rewriting 3,125 as a power of 5, then used exponent properties to simplify the equation. Finally, we isolated x and simplified the expression to find the value of x, which is 5.

Tips and Tricks

• When solving exponential equations, it's essential to use exponent properties to simplify the equation.
• Make sure to isolate the variable x on one side of the equation.
• Use division to isolate x, and simplify the expression to find the value of x.

Real-World Applications

Solving exponential equations has numerous real-world applications, including:

• Modeling population growth and decline
• Analyzing financial data and predicting stock prices
• Understanding chemical reactions and kinetics
• Solving problems in physics and engineering

Common Mistakes to Avoid

• Failing to use exponent properties to simplify the equation
• Not isolating x on one side of the equation
• Not simplifying the expression to find the value of x

Final Thoughts

Solving exponential equations is a fundamental aspect of mathematics that has numerous real-world applications. By using exponent properties and isolating x, we can find the value of x and solve complex equations. Remember to simplify the expression and avoid common mistakes to ensure accurate results.

Additional Resources

For more information on solving exponential equations, check out the following resources:

• Khan Academy: Exponential Equations
• Mathway: Exponential Equations
• Wolfram Alpha: Exponential Equations

Frequently Asked Questions

Q: What is the value of x in the equation $3,125 = 5^{-10 + 3x}$? A: The value of x is 5.

Q: How do I solve exponential equations? A: To solve exponential equations, use exponent properties to simplify the equation, isolate x, and simplify the expression to find the value of x.

Q: What are some real-world applications of solving exponential equations? A: Solving exponential equations has numerous real-world applications, including modeling population growth and decline, analyzing financial data, and understanding chemical reactions.

Introduction

Solving exponential equations is a fundamental aspect of mathematics that has numerous real-world applications. In our previous article, we solved the exponential equation $3,125 = 5^{-10 + 3x}$ for x. In this article, we will answer some frequently asked questions about solving exponential equations.

Q&A

Q: What is an exponential equation?

A: An exponential equation is an equation that involves a variable in the exponent of a power. For example, the equation $2^x = 8$ is an exponential equation.

Q: How do I solve exponential equations?

A: To solve exponential equations, use exponent properties to simplify the equation, isolate x, and simplify the expression to find the value of x.

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 use exponent properties to simplify the equation
• Not isolating x on one side of the equation
• Not simplifying the expression to find the value of x

Q: How do I use exponent properties to simplify exponential equations?

A: Exponent properties can be used to simplify exponential equations by:

• Using the product rule: $a^m \cdot a^n = a^{m+n}$
• Using the quotient rule: $\frac{a^m}{a^n} = a^{m-n}$
• Using the power rule: $(a^m)^n = a^{mn}$

Q: How do I isolate x in an exponential equation?

A: To isolate x in an exponential equation, you can use the following steps:

• Use exponent properties to simplify the equation
• Move all terms with x to one side of the equation
• Use division to isolate x

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

A: Solving exponential equations has numerous real-world applications, including:

• Modeling population growth and decline
• Analyzing financial data and predicting stock prices
• Understanding chemical reactions and kinetics
• Solving problems in physics and engineering

Q: How do I check my answer when solving an exponential equation?

A: To check your answer when solving an exponential equation, you can:

• Plug your answer back into the original equation
• Simplify the equation to see if it is true
• Use a calculator to check your answer

Q: What are some common types of exponential equations?

A: Some common types of exponential equations include:

• Equations with a base of 2: $2^x = 16$
• Equations with a base of 5: $5^x = 3125$
• Equations with a base of 10: $10^x = 1000$

Q: How do I solve exponential equations with a base of 2?

A: To solve exponential equations with a base of 2, you can use the following steps:

• Use exponent properties to simplify the equation
• Move all terms with x to one side of the equation
• Use division to isolate x

Q: How do I solve exponential equations with a base of 5?

A: To solve exponential equations with a base of 5, you can use the following steps:

• Use exponent properties to simplify the equation
• Move all terms with x to one side of the equation
• Use division to isolate x

Q: How do I solve exponential equations with a base of 10?

A: To solve exponential equations with a base of 10, you can use the following steps:

• Use exponent properties to simplify the equation
• Move all terms with x to one side of the equation
• Use division to isolate x

Conclusion

Solving exponential equations is a fundamental aspect of mathematics that has numerous real-world applications. By using exponent properties and isolating x, we can find the value of x and solve complex equations. Remember to simplify the expression and avoid common mistakes to ensure accurate results.

Additional Resources

For more information on solving exponential equations, check out the following resources:

• Khan Academy: Exponential Equations
• Mathway: Exponential Equations
• Wolfram Alpha: Exponential Equations

Frequently Asked Questions

Q: What is the value of x in the equation $3,125 = 5^{-10 + 3x}$? A: The value of x is 5.

Q: How do I solve exponential equations? A: To solve exponential equations, use exponent properties to simplify the equation, isolate x, and simplify the expression to find the value of x.

Q: What are some real-world applications of solving exponential equations? A: Solving exponential equations has numerous real-world applications, including modeling population growth and decline, analyzing financial data, and understanding chemical reactions.