The Expression $\log _3\left(\frac{9}{6}\right$\] Can Also Be Written As:

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Introduction

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Logarithmic expressions are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and computer science. In this article, we will explore the expression $\log _3\left(\frac{9}{6}\right)$ and simplify it using various techniques.

Understanding Logarithms

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A logarithm is the inverse operation of exponentiation. In other words, if $x$ is the logarithm of $y$ with base $b$, then $b^x = y$. For example, $\log _2(8) = 3$ because $2^3 = 8$. Logarithms are used to solve equations involving exponents and to simplify complex expressions.

Simplifying the Expression

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The given expression is $\log _3\left(\frac{9}{6}\right)$. To simplify this expression, we can use the properties of logarithms. One of the properties states that $\log _b\left(\frac{x}{y}\right) = \log _b(x) - \log _b(y)$. We can apply this property to the given expression.

Step 1: Apply the Quotient Rule

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Using the quotient rule, we can rewrite the expression as:

$$\log _3\left(\frac{9}{6}\right) = \log _3(9) - \log _3(6)$$

Step 2: Simplify the Logarithms

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Now, we can simplify the logarithms using the property that $\log _b(b^x) = x$. We can rewrite $\log _3(9)$ and $\log _3(6)$ as:

$$\log _3(9) = \log _3(3^2) = 2$$

$$\log _3(6) = \log _3(3 \cdot 2) = \log _3(3) + \log _3(2) = 1 + \log _3(2)$$

Step 3: Combine the Terms

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Now, we can combine the terms:

$$\log _3\left(\frac{9}{6}\right) = 2 - (1 + \log _3(2))$$

Step 4: Simplify the Expression

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Simplifying the expression further, we get:

$$\log _3\left(\frac{9}{6}\right) = 2 - 1 - \log _3(2)$$

$$\log _3\left(\frac{9}{6}\right) = 1 - \log _3(2)$$

Conclusion

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In this article, we simplified the expression $\log _3\left(\frac{9}{6}\right)$ using various techniques. We applied the quotient rule, simplified the logarithms, and combined the terms to get the final expression. The expression can be written as $1 - \log _3(2)$.

Final Answer

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The final answer is $\boxed{1 - \log _3(2)}$.

Related Topics

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• Logarithmic Properties: Learn about the properties of logarithms, including the quotient rule, product rule, and power rule.
• Exponentiation: Understand the concept of exponentiation and how it relates to logarithms.
• Simplifying Logarithmic Expressions: Learn how to simplify logarithmic expressions using various techniques.

References

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• Wikipedia: Logarithm
• Khan Academy: Logarithms
• Math Is Fun: Logarithms

Further Reading

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• Logarithmic Functions: Learn about logarithmic functions and their properties.
• Exponential Functions: Understand exponential functions and their properties.
• Calculus: Learn about calculus and how it relates to logarithms and exponentiation.
  

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Q: What is a logarithmic expression?

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A: A logarithmic expression is an expression that involves a logarithm, which is the inverse operation of exponentiation. In other words, if $x$ is the logarithm of $y$ with base $b$, then $b^x = y$.

Q: What are the properties of logarithms?

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A: The properties of logarithms include:

• Quotient Rule: $\log _b\left(\frac{x}{y}\right) = \log _b(x) - \log _b(y)$
• Product Rule: $\log _b(xy) = \log _b(x) + \log _b(y)$
• Power Rule: $\log _b(x^y) = y\log _b(x)$

Q: How do I simplify a logarithmic expression?

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A: To simplify a logarithmic expression, you can use the properties of logarithms. Here are the steps:

1. Apply the Quotient Rule: If the expression involves a fraction, apply the quotient rule to rewrite it as the difference of two logarithms.
2. Simplify the Logarithms: Use the property that $\log _b(b^x) = x$ to simplify the logarithms.
3. Combine the Terms: Combine the terms using the product rule and power rule.

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

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A: A logarithmic expression is an expression that involves a logarithm, while an exponential expression is an expression that involves an exponent. For example, $\log _3(9)$ is a logarithmic expression, while $3^2$ is an exponential expression.

Q: How do I evaluate a logarithmic expression?

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A: To evaluate a logarithmic expression, you need to find the value of the logarithm. Here are the steps:

1. Understand the Base: Understand the base of the logarithm, which is the number that is being raised to the power.
2. Understand the Argument: Understand the argument of the logarithm, which is the number that is being logged.
3. Use a Calculator: Use a calculator to find the value of the logarithm.

Q: What are some common logarithmic expressions?

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A: Some common logarithmic expressions include:

• $\log _b(b)$: This is a logarithmic expression that evaluates to 1.
• $\log _b(1)$: This is a logarithmic expression that evaluates to 0.
• $\log _b(b^x)$: This is a logarithmic expression that evaluates to $x$.

Q: How do I graph a logarithmic function?

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A: To graph a logarithmic function, you need to understand the properties of the function. Here are the steps:

1. Understand the Base: Understand the base of the logarithm, which is the number that is being raised to the power.
2. Understand the Argument: Understand the argument of the logarithm, which is the number that is being logged.
3. Use a Graphing Calculator: Use a graphing calculator to graph the function.

Q: What are some real-world applications of logarithmic expressions?

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A: Logarithmic expressions have many real-world applications, including:

• Finance: Logarithmic expressions are used to calculate interest rates and investment returns.
• Science: Logarithmic expressions are used to calculate the pH of a solution and the concentration of a substance.
• Engineering: Logarithmic expressions are used to calculate the gain of an amplifier and the frequency response of a circuit.

Q: How do I solve a logarithmic equation?

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A: To solve a logarithmic equation, you need to isolate the logarithmic expression. Here are the steps:

1. Understand the Equation: Understand the equation and the properties of the logarithm.
2. Isolate the Logarithmic Expression: Isolate the logarithmic expression using algebraic manipulations.
3. Use a Calculator: Use a calculator to find the value of the logarithm.

Q: What are some common mistakes to avoid when working with logarithmic expressions?

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A: Some common mistakes to avoid when working with logarithmic expressions include:

• Confusing the Base and the Argument: Make sure to understand the base and the argument of the logarithm.
• Forgetting to Use the Properties of Logarithms: Make sure to use the properties of logarithms to simplify the expression.
• Not Checking the Domain: Make sure to check the domain of the logarithmic expression to avoid errors.