Simplify The Expression:${ \frac{1}{2} \log_{10} 36 - \log_{10} 15 + 2 \log_{10} 5 }$

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Introduction

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Logarithmic equations are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will focus on simplifying the expression $\frac{1}{2} \log_{10} 36 - \log_{10} 15 + 2 \log_{10} 5$. We will break down the expression into manageable parts, apply logarithmic properties, and arrive at a simplified solution.

Understanding Logarithmic Properties

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Before we dive into the simplification process, it's essential to understand the basic logarithmic properties. These properties will help us manipulate the given expression and arrive at a simplified solution.

• Product Property: $\log_{a} (xy) = \log_{a} x + \log_{a} y$
• Quotient Property: $\log_{a} \frac{x}{y} = \log_{a} x - \log_{a} y$
• Power Property: $\log_{a} x^{y} = y \log_{a} x$

Breaking Down the Expression

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The given expression is $\frac{1}{2} \log_{10} 36 - \log_{10} 15 + 2 \log_{10} 5$. To simplify this expression, we need to break it down into manageable parts.

Part 1: Simplifying $\frac{1}{2} \log_{10} 36$

We can start by simplifying the first part of the expression, $\frac{1}{2} \log_{10} 36$. Using the power property, we can rewrite this as $\log_{10} 36^{\frac{1}{2}}$.

$\log_{10} 36^{\frac{1}{2}} = \log_{10} \sqrt{36}$

Since $\sqrt{36} = 6$, we can further simplify this as:

$\log_{10} 6$

Part 2: Simplifying $\log_{10} 15$

The second part of the expression is $\log_{10} 15$. This term remains unchanged for now.

Part 3: Simplifying $2 \log_{10} 5$

The third part of the expression is $2 \log_{10} 5$. Using the power property, we can rewrite this as $\log_{10} 5^{2}$.

$\log_{10} 5^{2} = \log_{10} 25$

Combining the Parts

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Now that we have simplified each part of the expression, we can combine them to arrive at the final solution.

$\log_{10} 6 - \log_{10} 15 + \log_{10} 25$

Applying Logarithmic Properties

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We can further simplify the expression by applying logarithmic properties. Using the product property, we can rewrite the expression as:

$\log_{10} (6 \cdot 25) - \log_{10} 15$

$\log_{10} 150 - \log_{10} 15$

Simplifying the Expression

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The final step is to simplify the expression by combining the logarithmic terms.

$\log_{10} 150 - \log_{10} 15 = \log_{10} \frac{150}{15}$

$\log_{10} \frac{150}{15} = \log_{10} 10$

Conclusion

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In this article, we simplified the expression $\frac{1}{2} \log_{10} 36 - \log_{10} 15 + 2 \log_{10} 5$ by breaking it down into manageable parts, applying logarithmic properties, and combining the terms. The final solution is $\log_{10} 10$, which is equal to $1$.

Frequently Asked Questions

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Q: What is the product property of logarithms?

A: The product property of logarithms states that $\log_{a} (xy) = \log_{a} x + \log_{a} y$.

Q: What is the quotient property of logarithms?

A: The quotient property of logarithms states that $\log_{a} \frac{x}{y} = \log_{a} x - \log_{a} y$.

Q: What is the power property of logarithms?

A: The power property of logarithms states that $\log_{a} x^{y} = y \log_{a} x$.

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, you can break it down into manageable parts, apply logarithmic properties, and combine the terms.

Q: What is the final solution to the expression $\frac{1}{2} \log_{10} 36 - \log_{10} 15 + 2 \log_{10} 5$?

A: The final solution to the expression $\frac{1}{2} \log_{10} 36 - \log_{10} 15 + 2 \log_{10} 5$ is $\log_{10} 10$, which is equal to $1$.

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Introduction

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Logarithmic equations are a fundamental concept in mathematics, and understanding them is essential for students and professionals alike. In this article, we will provide a comprehensive guide to logarithmic equations, including a Q&A section that covers common questions and topics.

Understanding Logarithmic Equations

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Logarithmic equations are a way of expressing exponential relationships in a different form. They are used to solve problems involving growth and decay, and are a fundamental concept in mathematics.

What is a Logarithmic Equation?

A logarithmic equation is an equation that involves a logarithm, which is the inverse of an exponential function. Logarithmic equations are used to solve problems involving growth and decay, and are a fundamental concept in mathematics.

What is the Difference Between Logarithmic and Exponential Equations?

Logarithmic equations and exponential equations are related, but they are not the same thing. Exponential equations involve exponential functions, while logarithmic equations involve logarithms.

What is the Product Property of Logarithms?

The product property of logarithms states that $\log_{a} (xy) = \log_{a} x + \log_{a} y$. This property allows us to simplify logarithmic expressions by combining the logarithms of multiple numbers.

What is the Quotient Property of Logarithms?

The quotient property of logarithms states that $\log_{a} \frac{x}{y} = \log_{a} x - \log_{a} y$. This property allows us to simplify logarithmic expressions by combining the logarithms of multiple numbers.

What is the Power Property of Logarithms?

The power property of logarithms states that $\log_{a} x^{y} = y \log_{a} x$. This property allows us to simplify logarithmic expressions by combining the logarithms of multiple numbers.

Q&A

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

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse of an exponential function. An exponential equation, on the other hand, involves an exponential function.

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, you can use the product property, quotient property, and power property of logarithms.

Q: What is the product property of logarithms?

A: The product property of logarithms states that $\log_{a} (xy) = \log_{a} x + \log_{a} y$.

Q: What is the quotient property of logarithms?

A: The quotient property of logarithms states that $\log_{a} \frac{x}{y} = \log_{a} x - \log_{a} y$.

Q: What is the power property of logarithms?

A: The power property of logarithms states that $\log_{a} x^{y} = y \log_{a} x$.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the properties of logarithms to simplify the equation and then solve for the variable.

Q: What is the inverse of a logarithmic function?

A: The inverse of a logarithmic function is an exponential function.

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you can use a graphing calculator or graph paper.

Q: What is the domain of a logarithmic function?

A: The domain of a logarithmic function is all real numbers greater than zero.

Q: What is the range of a logarithmic function?

A: The range of a logarithmic function is all real numbers.

Conclusion

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In this article, we have provided a comprehensive guide to logarithmic equations, including a Q&A section that covers common questions and topics. We have discussed the properties of logarithms, how to simplify logarithmic expressions, and how to solve logarithmic equations. We have also covered common questions and topics related to logarithmic equations.

Frequently Asked Questions

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

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse of an exponential function. An exponential equation, on the other hand, involves an exponential function.

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, you can use the product property, quotient property, and power property of logarithms.

Q: What is the product property of logarithms?

A: The product property of logarithms states that $\log_{a} (xy) = \log_{a} x + \log_{a} y$.

Q: What is the quotient property of logarithms?

A: The quotient property of logarithms states that $\log_{a} \frac{x}{y} = \log_{a} x - \log_{a} y$.

Q: What is the power property of logarithms?

A: The power property of logarithms states that $\log_{a} x^{y} = y \log_{a} x$.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the properties of logarithms to simplify the equation and then solve for the variable.

Q: What is the inverse of a logarithmic function?

A: The inverse of a logarithmic function is an exponential function.

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you can use a graphing calculator or graph paper.

Q: What is the domain of a logarithmic function?

A: The domain of a logarithmic function is all real numbers greater than zero.

Q: What is the range of a logarithmic function?

A: The range of a logarithmic function is all real numbers.