Solve For { X $} : : : { 5.125^{x+3} = \frac{1}{25} \}

Introduction

Exponential equations can be challenging to solve, especially when dealing with variables in the exponent. In this article, we will focus on solving the equation $5.125^{x+3} = \frac{1}{25}$ to find the value of $x$. We will break down the solution into manageable steps, using algebraic manipulations and properties of exponents to isolate the variable.

Understanding the Equation

The given equation is $5.125^{x+3} = \frac{1}{25}$. To begin solving this equation, we need to understand the properties of exponents and how to manipulate them. The base of the exponent is $5.125$, and the exponent is $x+3$. The right-hand side of the equation is $\frac{1}{25}$, which can be rewritten as $5^{-2}$.

Step 1: Rewrite the Equation with a Common Base

To make it easier to compare the two sides of the equation, we can rewrite the right-hand side with a common base. Since the base of the left-hand side is $5.125$, we can rewrite the right-hand side as $5^{-2}$. This gives us the equation:

$$5.125^{x+3} = 5^{-2}$$

Step 2: Use the Property of Exponents to Simplify the Equation

Now that we have a common base, we can use the property of exponents that states $a^m = a^n$ implies $m = n$. However, in this case, we have different bases, so we cannot directly compare the exponents. Instead, we can use the property that states $a^m = a^n$ implies $\log_a a^m = \log_a a^n$, which simplifies to $m = n$. We can apply this property to our equation by taking the logarithm of both sides.

Step 3: Take the Logarithm of Both Sides

To eliminate the exponent, we can take the logarithm of both sides of the equation. We will use the logarithm base 10, but we can use any base. Taking the logarithm of both sides gives us:

$$\log_{10} 5.125^{x+3} = \log_{10} 5^{-2}$$

Step 4: Use the Property of Logarithms to Simplify the Equation

Now that we have taken the logarithm of both sides, we can use the property of logarithms that states $\log_a b^c = c \log_a b$. Applying this property to our equation gives us:

$$(x+3) \log_{10} 5.125 = -2 \log_{10} 5$$

Step 5: Simplify the Equation

Now that we have simplified the equation, we can isolate the variable $x$. To do this, we can divide both sides of the equation by $\log_{10} 5.125$.

$$(x+3) = \frac{-2 \log_{10} 5}{\log_{10} 5.125}$$

Step 6: Solve for x

Now that we have isolated the variable $x$, we can solve for its value. To do this, we can subtract 3 from both sides of the equation.

$$x = \frac{-2 \log_{10} 5}{\log_{10} 5.125} - 3$$

Conclusion

In this article, we have solved the exponential equation $5.125^{x+3} = \frac{1}{25}$ to find the value of $x$. We used algebraic manipulations and properties of exponents to isolate the variable. The final solution is $x = \frac{-2 \log_{10} 5}{\log_{10} 5.125} - 3$. This solution demonstrates the importance of understanding the properties of exponents and how to manipulate them to solve complex equations.

Calculating the Value of x

To calculate the value of $x$, we can use a calculator to evaluate the expression $\frac{-2 \log_{10} 5}{\log_{10} 5.125} - 3$. Plugging in the values, we get:

$$x = \frac{-2 \log_{10} 5}{\log_{10} 5.125} - 3 \approx \frac{-2 \times -0.69897}{0.69466} - 3 \approx 2.016 - 3 \approx -0.984$$

Therefore, the value of $x$ is approximately $-0.984$.

Final Answer

The final answer is $\boxed{-0.984}$.

Introduction

In our previous article, we solved the exponential equation $5.125^{x+3} = \frac{1}{25}$ to find the value of $x$. In this article, we will provide a Q&A guide to help you understand the solution and answer any questions you may have.

Q: What is an exponential equation?

A: An exponential equation is an equation that contains a variable in the exponent. For example, $5^{x+3} = 10$ is an exponential equation.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable in the exponent. You can do this by using algebraic manipulations and properties of exponents.

Q: What is the property of exponents that states $a^m = a^n$ implies $m = n$?

A: This property is called the "one-to-one" property of exponents. It states that if $a^m = a^n$, then $m = n$. However, this property only applies when the bases are the same.

Q: How do I take the logarithm of both sides of an equation?

A: To take the logarithm of both sides of an equation, you need to use the logarithm base that is the same as the base of the exponent. For example, if the equation is $5^{x+3} = 10$, you would take the logarithm base 10 of both sides.

Q: What is the property of logarithms that states $\log_a b^c = c \log_a b$?

A: This property is called the "power rule" of logarithms. It states that if $b^c = a$, then $\log_a b^c = c \log_a b$.

Q: How do I simplify an equation with logarithms?

A: To simplify an equation with logarithms, you can use the properties of logarithms to combine the logarithms and simplify the equation.

Q: What is the final solution to the equation $5.125^{x+3} = \frac{1}{25}$?

A: The final solution to the equation $5.125^{x+3} = \frac{1}{25}$ is $x = \frac{-2 \log_{10} 5}{\log_{10} 5.125} - 3$.

Q: How do I calculate the value of $x$?

A: To calculate the value of $x$, you can use a calculator to evaluate the expression $\frac{-2 \log_{10} 5}{\log_{10} 5.125} - 3$.

Q: What is the value of $x$?

A: The value of $x$ is approximately $-0.984$.

Q: Can I use this method to solve other exponential equations?

A: Yes, you can use this method to solve other exponential equations. The key is to isolate the variable in the exponent and use the properties of exponents and logarithms to simplify the equation.

Conclusion

In this Q&A guide, we have provided answers to common questions about solving exponential equations. We have also provided a step-by-step guide to solving the equation $5.125^{x+3} = \frac{1}{25}$. By following this guide, you should be able to solve other exponential equations and understand the properties of exponents and logarithms.

Additional Resources

• For more information on solving exponential equations, see our previous article "Solving Exponential Equations: A Step-by-Step Guide".
• For more information on the properties of exponents and logarithms, see our article "Exponents and Logarithms: A Guide to Understanding the Basics".
• For practice problems and exercises, see our article "Exponential Equations: Practice Problems and Exercises".