If Log ⁡ 10 Y + 3 Log ⁡ 10 X = 2 \log_{10} Y + 3 \log_{10} X = 2 Lo G 10 ​ Y + 3 Lo G 10 ​ X = 2 , Express Y Y Y In Terms Of X X X .

Introduction

In this article, we will delve into the world of logarithms and explore a mathematical problem that requires us to express a variable in terms of another variable. The problem at hand is to express $y$ in terms of $x$ given the equation $\log_{10} y + 3 \log_{10} x = 2$. We will use various mathematical techniques and properties of logarithms to derive the expression for $y$ in terms of $x$.

Understanding Logarithms

Before we proceed with the problem, let's take a moment to understand the concept of logarithms. A logarithm is the inverse operation of exponentiation. In other words, if $a^b = c$, then $\log_a c = b$. The logarithm of a number to a certain base is the exponent to which the base must be raised to produce that number. For example, $\log_{10} 100 = 2$ because $10^2 = 100$.

Using Properties of Logarithms

To solve the problem, we will use the properties of logarithms. One of the most important properties of logarithms is the product rule, which states that $\log_a (m \cdot n) = \log_a m + \log_a n$. We will also use the power rule, which states that $\log_a (m^b) = b \cdot \log_a m$.

Deriving the Expression for y

Now that we have a good understanding of logarithms and their properties, let's proceed with the problem. We are given the equation $\log_{10} y + 3 \log_{10} x = 2$. Our goal is to express $y$ in terms of $x$.

Using the power rule, we can rewrite the equation as $\log_{10} y + \log_{10} x^3 = 2$. This simplifies to $\log_{10} y + \log_{10} (x^3) = 2$.

Next, we can use the product rule to combine the two logarithms on the left-hand side of the equation. This gives us $\log_{10} (y \cdot x^3) = 2$.

Now, we can use the definition of logarithms to rewrite the equation in exponential form. This gives us $y \cdot x^3 = 10^2$.

Simplifying the right-hand side of the equation, we get $y \cdot x^3 = 100$.

Finally, we can solve for $y$ by dividing both sides of the equation by $x^3$. This gives us $y = \frac{100}{x^3}$.

Conclusion

In this article, we have derived the expression for $y$ in terms of $x$ given the equation $\log_{10} y + 3 \log_{10} x = 2$. We used various mathematical techniques and properties of logarithms to solve the problem. The final expression for $y$ in terms of $x$ is $y = \frac{100}{x^3}$.

Example Use Cases

The expression for $y$ in terms of $x$ can be used in a variety of mathematical and real-world applications. For example, it can be used to model the relationship between two variables in a scientific experiment or to solve a system of equations.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

1. Use the power rule to rewrite the equation as $\log_{10} y + \log_{10} x^3 = 2$.
2. Use the product rule to combine the two logarithms on the left-hand side of the equation. This gives us $\log_{10} (y \cdot x^3) = 2$.
3. Use the definition of logarithms to rewrite the equation in exponential form. This gives us $y \cdot x^3 = 10^2$.
4. Simplify the right-hand side of the equation to get $y \cdot x^3 = 100$.
5. Solve for $y$ by dividing both sides of the equation by $x^3$. This gives us $y = \frac{100}{x^3}$.

Mathematical Derivations

Here are the mathematical derivations for the properties of logarithms used in the solution:

• Product Rule: $\log_a (m \cdot n) = \log_a m + \log_a n$
• Power Rule: $\log_a (m^b) = b \cdot \log_a m$

Real-World Applications

The expression for $y$ in terms of $x$ can be used in a variety of real-world applications, such as:

• Scientific Experiments: The expression can be used to model the relationship between two variables in a scientific experiment.
• Engineering: The expression can be used to solve a system of equations in engineering applications.
• Economics: The expression can be used to model the relationship between two variables in economic models.

Conclusion

Introduction

In our previous article, we derived the expression for $y$ in terms of $x$ given the equation $\log_{10} y + 3 \log_{10} x = 2$. We used various mathematical techniques and properties of logarithms to solve the problem. In this article, we will answer some of the most frequently asked questions related to the problem.

Q: What is the final expression for y in terms of x?

A: The final expression for $y$ in terms of $x$ is $y = \frac{100}{x^3}$.

Q: How did you derive the expression for y in terms of x?

A: We used the power rule and the product rule of logarithms to derive the expression for $y$ in terms of $x$. We started with the equation $\log_{10} y + 3 \log_{10} x = 2$ and used the power rule to rewrite it as $\log_{10} y + \log_{10} x^3 = 2$. We then used the product rule to combine the two logarithms on the left-hand side of the equation, which gave us $\log_{10} (y \cdot x^3) = 2$. We then used the definition of logarithms to rewrite the equation in exponential form, which gave us $y \cdot x^3 = 10^2$. We then simplified the right-hand side of the equation to get $y \cdot x^3 = 100$. Finally, we solved for $y$ by dividing both sides of the equation by $x^3$, which gave us $y = \frac{100}{x^3}$.

Q: What are some of the real-world applications of the expression for y in terms of x?

A: The expression for $y$ in terms of $x$ can be used in a variety of real-world applications, such as:

• Scientific Experiments: The expression can be used to model the relationship between two variables in a scientific experiment.
• Engineering: The expression can be used to solve a system of equations in engineering applications.
• Economics: The expression can be used to model the relationship between two variables in economic models.

Q: What are some of the mathematical techniques used to derive the expression for y in terms of x?

A: We used the following mathematical techniques to derive the expression for $y$ in terms of $x$:

• Power Rule: $\log_a (m^b) = b \cdot \log_a m$
• Product Rule: $\log_a (m \cdot n) = \log_a m + \log_a n$
• Definition of Logarithms: $a^b = c \implies \log_a c = b$

Q: Can you provide a step-by-step solution to the problem?

A: Here is a step-by-step solution to the problem:

1. Use the power rule to rewrite the equation as $\log_{10} y + \log_{10} x^3 = 2$.
2. Use the product rule to combine the two logarithms on the left-hand side of the equation. This gives us $\log_{10} (y \cdot x^3) = 2$.
3. Use the definition of logarithms to rewrite the equation in exponential form. This gives us $y \cdot x^3 = 10^2$.
4. Simplify the right-hand side of the equation to get $y \cdot x^3 = 100$.
5. Solve for $y$ by dividing both sides of the equation by $x^3$. This gives us $y = \frac{100}{x^3}$.

Q: What are some of the common mistakes to avoid when deriving the expression for y in terms of x?

A: Some of the common mistakes to avoid when deriving the expression for $y$ in terms of $x$ include:

• Forgetting to use the power rule: Make sure to use the power rule to rewrite the equation as $\log_{10} y + \log_{10} x^3 = 2$.
• Forgetting to use the product rule: Make sure to use the product rule to combine the two logarithms on the left-hand side of the equation.
• Forgetting to simplify the right-hand side of the equation: Make sure to simplify the right-hand side of the equation to get $y \cdot x^3 = 100$.
• Forgetting to solve for y: Make sure to solve for $y$ by dividing both sides of the equation by $x^3$.

Conclusion

In this article, we have answered some of the most frequently asked questions related to the problem of expressing $y$ in terms of $x$ given the equation $\log_{10} y + 3 \log_{10} x = 2$. We have provided a step-by-step solution to the problem and highlighted some of the common mistakes to avoid when deriving the expression for $y$ in terms of $x$.