Find \[$ X \$\] In The Following Equation:$\[ \log_{10}(x+3) - \log_{10}(x-3) = 1 \\]

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Introduction

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Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. In this article, we will focus on finding the value of ${ x }$ in the given logarithmic equation: ${ \log_{10}(x+3) - \log_{10}(x-3) = 1 }$

Understanding Logarithmic Properties

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Before we dive into solving the equation, it's essential to understand the properties of logarithms. The logarithmic function is the inverse of the exponential function, and it's defined as the power to which a base number must be raised to produce a given value. In this case, we're dealing with a logarithm with base 10.

One of the key properties of logarithms is the product rule, which states that $\log_{b}(xy) = \log_{b}(x) + \log_{b}(y)$. Another important property is the quotient rule, which states that $\log_{b}\left(\frac{x}{y}\right) = \log_{b}(x) - \log_{b}(y)$.

Applying the Quotient Rule

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In the given equation, we have two logarithmic terms with the same base: $\log_{10}(x+3)$ and $\log_{10}(x-3)$. We can use the quotient rule to simplify the equation by combining the two logarithmic terms into a single logarithmic term.

Using the quotient rule, we can rewrite the equation as: $\log_{10}\left(\frac{x+3}{x-3}\right) = 1$

Exponentiating Both Sides

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To solve for ${ x }$, we need to get rid of the logarithm. We can do this by exponentiating both sides of the equation. Since the base of the logarithm is 10, we can use the fact that $10^{\log_{10}(x)} = x$.

Exponentiating both sides of the equation, we get: $\frac{x+3}{x-3} = 10^1$

Simplifying the Equation

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We can simplify the equation by multiplying both sides by $x-3$: $x+3 = 10(x-3)$

Expanding and Simplifying

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Expanding the right-hand side of the equation, we get: $x+3 = 10x - 30$

Solving for ${ x }$

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Subtracting $x$ from both sides of the equation, we get: $3 = 9x - 30$

Adding 30 to both sides of the equation, we get: $33 = 9x$

Dividing both sides of the equation by 9, we get: $x = \frac{33}{9}$

Simplifying the Fraction

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We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This gives us: $x = \frac{11}{3}$

Conclusion

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In this article, we solved the logarithmic equation ${ \log_{10}(x+3) - \log_{10}(x-3) = 1 }$ by applying the quotient rule, exponentiating both sides, and simplifying the resulting equation. We found that the value of ${ x }$ is $\boxed{\frac{11}{3}}$.

Final Thoughts

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Solving logarithmic equations requires a deep understanding of the properties of logarithms and a step-by-step approach. By following the steps outlined in this article, you can solve logarithmic equations and gain a deeper understanding of this fundamental concept in mathematics.

Additional Resources

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If you're interested in learning more about logarithmic equations, here are some additional resources:

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

Frequently Asked Questions

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Q: What is the definition of a logarithm?

A: A logarithm is the power to which a base number must be raised to produce a given value.

Q: What is the quotient rule for logarithms?

A: The quotient rule for logarithms states that $\log_{b}\left(\frac{x}{y}\right) = \log_{b}(x) - \log_{b}(y)$.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can apply the quotient rule, exponentiate both sides, and simplify the resulting equation.

Q: What is the value of ${ x }$ in the given equation?

A: The value of ${ x }$ in the given equation is $\boxed{\frac{11}{3}}$.

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Introduction

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Logarithmic equations can be challenging to solve, but with the right approach and understanding of the properties of logarithms, you can tackle even the most complex equations. In this article, we'll answer some of the most frequently asked questions about logarithmic equations.

Q&A

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Q: What is the definition of a logarithm?

A: A logarithm is the power to which a base number must be raised to produce a given value. For example, $\log_{10}(100) = 2$ because $10^2 = 100$.

Q: What is the quotient rule for logarithms?

A: The quotient rule for logarithms states that $\log_{b}\left(\frac{x}{y}\right) = \log_{b}(x) - \log_{b}(y)$. This rule allows us to simplify complex logarithmic expressions by combining them into a single logarithmic term.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can follow these steps:

1. Apply the quotient rule to simplify the equation.
2. Exponentiate both sides of the equation to get rid of the logarithm.
3. Simplify the resulting equation to solve for the variable.

Q: What is the difference between a logarithmic equation and an exponential equation?

A: A logarithmic equation is an equation that involves a logarithmic function, while an exponential equation is an equation that involves an exponential function. For example, $\log_{10}(x) = 2$ is a logarithmic equation, while $10^x = 100$ is an exponential equation.

Q: Can I use a calculator to solve logarithmic equations?

A: Yes, you can use a calculator to solve logarithmic equations. However, it's essential to understand the underlying math and be able to apply the properties of logarithms to solve the equation.

Q: What is the value of ${ x }$ in the equation $\log_{10}(x) = 3$?

A: To solve this equation, we can exponentiate both sides to get $x = 10^3 = 1000$.

Q: What is the value of ${ x }$ in the equation $\log_{10}(x) = -2$?

A: To solve this equation, we can exponentiate both sides to get $x = 10^{-2} = 0.01$.

Q: Can I use logarithmic equations to solve real-world problems?

A: Yes, logarithmic equations can be used to solve a wide range of real-world problems, including problems involving population growth, chemical reactions, and financial calculations.

Q: What are some common applications of logarithmic equations?

A: Logarithmic equations have many applications in fields such as:

• Biology: to model population growth and decay
• Chemistry: to model chemical reactions and equilibrium
• Finance: to calculate interest rates and investment returns
• Physics: to model sound waves and light waves

Conclusion

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Logarithmic equations can be challenging to solve, but with the right approach and understanding of the properties of logarithms, you can tackle even the most complex equations. By following the steps outlined in this article and practicing with real-world examples, you can become proficient in solving logarithmic equations and apply them to a wide range of fields.

Additional Resources

---

If you're interested in learning more about logarithmic equations, here are some additional resources:

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

Frequently Asked Questions

---

Q: What is the definition of a logarithm?

A: A logarithm is the power to which a base number must be raised to produce a given value.

Q: What is the quotient rule for logarithms?

A: The quotient rule for logarithms states that $\log_{b}\left(\frac{x}{y}\right) = \log_{b}(x) - \log_{b}(y)$.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can apply the quotient rule, exponentiate both sides, and simplify the resulting equation.

Q: What is the value of ${ x }$ in the given equation?

A: The value of ${ x }$ in the given equation is $\boxed{\frac{11}{3}}$.

Q: Can I use a calculator to solve logarithmic equations?

A: Yes, you can use a calculator to solve logarithmic equations. However, it's essential to understand the underlying math and be able to apply the properties of logarithms to solve the equation.

Q: What are some common applications of logarithmic equations?

A: Logarithmic equations have many applications in fields such as biology, chemistry, finance, and physics.