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

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Introduction

Mathematics is a vast and fascinating subject that encompasses various branches, including algebra, geometry, and calculus. One of the fundamental concepts in 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 a specific equation involving exponents and logarithms.

Understanding the 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 equation involves an exponent, which can be simplified using logarithmic properties.

Simplifying the Equation

The first step in solving the equation is to simplify the right-hand side by applying the properties of exponents. We can rewrite the equation as:

$$3,125 = 5^{-10} \cdot 5^{3x}$$

Applying Logarithmic Properties

To simplify the equation further, we can apply the logarithmic properties. We can take the logarithm of both sides of the equation, which will help us to isolate the variable $x$. We can use the logarithmic property $\log(a \cdot b) = \log(a) + \log(b)$ to simplify the equation.

$$\log(3,125) = \log(5^{-10}) + \log(5^{3x})$$

Using Logarithmic Tables

To simplify the equation further, we can use logarithmic tables. We can look up the logarithm of 3,125 and the logarithm of 5 to the power of -10.

$$\log(3,125) = 3.49$$

$$\log(5^{-10}) = -10 \cdot \log(5) = -10 \cdot 0.70 = -7.00$$

Isolating the Variable

Now that we have simplified the equation, we can isolate the variable $x$ by applying the logarithmic property $\log(a^b) = b \cdot \log(a)$. We can rewrite the equation as:

$$3.49 = -7.00 + 3x \cdot \log(5)$$

Solving for $x$

To solve for $x$, we can add 7.00 to both sides of the equation and then divide both sides by $3 \cdot \log(5)$.

$$3.49 + 7.00 = 3x \cdot \log(5)$$

$$10.49 = 3x \cdot \log(5)$$

$$x = \frac{10.49}{3 \cdot \log(5)}$$

Calculating the Value of $x$

To calculate the value of $x$, we can use a calculator to evaluate the expression $\frac{10.49}{3 \cdot \log(5)}$.

$$x = \frac{10.49}{3 \cdot 0.70}$$

$$x = \frac{10.49}{2.10}$$

$$x = 5.00$$

Conclusion

In this article, we have solved the equation $3,125 = 5^{-10 + 3x}$ for $x$. We have applied logarithmic properties and used logarithmic tables to simplify the equation and isolate the variable $x$. The final value of $x$ is 5.00.

Frequently Asked Questions

• Q: What is the value of $x$ in the equation $3,125 = 5^{-10 + 3x}$? A: The value of $x$ is 5.00.
• Q: How do we simplify the equation $3,125 = 5^{-10 + 3x}$? A: We can simplify the equation by applying logarithmic properties and using logarithmic tables.
• Q: What is the final value of $x$ in the equation $3,125 = 5^{-10 + 3x}$? A: The final value of $x$ is 5.00.

References

• [1] "Logarithmic Properties" by Math Open Reference
• [2] "Logarithmic Tables" by Wolfram Alpha
• [3] "Solving Equations" by Khan Academy
  

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Introduction

In our previous article, we solved the equation $3,125 = 5^{-10 + 3x}$ for $x$. We applied logarithmic properties and used logarithmic tables to simplify the equation and isolate the variable $x$. In this article, we will provide a Q&A section to answer some of the frequently asked questions related to the equation.

Q&A

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

A: The value of $x$ is 5.00.

Q: How do we simplify the equation $3,125 = 5^{-10 + 3x}$?

A: We can simplify the equation by applying logarithmic properties and using logarithmic tables.

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

A: The final value of $x$ is 5.00.

Q: Can we solve the equation $3,125 = 5^{-10 + 3x}$ using other methods?

A: Yes, we can solve the equation using other methods such as substitution or elimination. However, the method we used in our previous article is the most efficient way to solve the equation.

Q: What is the significance of the logarithmic properties in solving the equation $3,125 = 5^{-10 + 3x}$?

A: The logarithmic properties are essential in solving the equation because they allow us to simplify the equation and isolate the variable $x$. Without the logarithmic properties, we would not be able to solve the equation.

Q: Can we use logarithmic tables to solve other equations?

A: Yes, we can use logarithmic tables to solve other equations that involve exponents and logarithms. However, we need to make sure that the equation is in the correct form and that we are using the correct logarithmic properties.

Q: What is the relationship between the logarithmic properties and the equation $3,125 = 5^{-10 + 3x}$?

A: The logarithmic properties are used to simplify the equation and isolate the variable $x$. The equation $3,125 = 5^{-10 + 3x}$ is a classic example of how logarithmic properties can be used to solve equations involving exponents and logarithms.

Q: Can we use a calculator to solve the equation $3,125 = 5^{-10 + 3x}$?

A: Yes, we can use a calculator to solve the equation. However, we need to make sure that we are using the correct logarithmic properties and that we are entering the correct values into the calculator.

Q: What is the final answer to the equation $3,125 = 5^{-10 + 3x}$?

A: The final answer to the equation is $x = 5.00$.

Conclusion

In this article, we have provided a Q&A section to answer some of the frequently asked questions related to the equation $3,125 = 5^{-10 + 3x}$. We have discussed the significance of the logarithmic properties in solving the equation and provided examples of how to use logarithmic tables to solve other 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 5.00.
• Q: How do we simplify the equation $3,125 = 5^{-10 + 3x}$? A: We can simplify the equation by applying logarithmic properties and using logarithmic tables.
• Q: What is the final value of $x$ in the equation $3,125 = 5^{-10 + 3x}$? A: The final value of $x$ is 5.00.

References

• [1] "Logarithmic Properties" by Math Open Reference
• [2] "Logarithmic Tables" by Wolfram Alpha
• [3] "Solving Equations" by Khan Academy