Write The Expression As A Single Logarithm.${ 6 \log_7 X - \frac{1}{5} \log_7 Y + 3 \log_7 Z }$

by ADMIN 97 views

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and they play a crucial role in various fields, including physics, engineering, and computer science. In this article, we will focus on simplifying a given logarithmic expression into a single logarithm. We will use the properties of logarithms to rewrite the expression in a more compact and manageable form.

The Properties of Logarithms

Before we dive into the simplification process, let's review the properties of logarithms. The logarithm of a number is the exponent to which a base must be raised to produce that number. In mathematical notation, this is represented as:

log_b(x) = y

This means that b^y = x.

There are several properties of logarithms that we will use to simplify the given expression:

  • Product Property: log_b(xy) = log_b(x) + log_b(y)
  • Quotient Property: log_b(x/y) = log_b(x) - log_b(y)
  • Power Property: log_b(x^y) = y * log_b(x)

Simplifying the Given Expression

Now that we have reviewed the properties of logarithms, let's simplify the given expression:

6log⁡7x−15log⁡7y+3log⁡7z{ 6 \log_7 x - \frac{1}{5} \log_7 y + 3 \log_7 z }

To simplify this expression, we will use the properties of logarithms to combine the terms.

First, let's rewrite the expression using the product property:

6log⁡7x−15log⁡7y+3log⁡7z=log⁡7(x6)−log⁡7(y15)+log⁡7(z3){ 6 \log_7 x - \frac{1}{5} \log_7 y + 3 \log_7 z = \log_7 (x^6) - \log_7 (y^{\frac{1}{5}}) + \log_7 (z^3) }

Next, let's use the quotient property to combine the first two terms:

log⁡7(x6)−log⁡7(y15)=log⁡7(x6y15){ \log_7 (x^6) - \log_7 (y^{\frac{1}{5}}) = \log_7 \left( \frac{x^6}{y^{\frac{1}{5}}} \right) }

Now, let's use the power property to rewrite the third term:

log⁡7(z3)=3log⁡7z{ \log_7 (z^3) = 3 \log_7 z }

Finally, let's use the product property to combine the first two terms:

log⁡7(x6y15)+3log⁡7z=log⁡7(x6y15⋅z3){ \log_7 \left( \frac{x^6}{y^{\frac{1}{5}}} \right) + 3 \log_7 z = \log_7 \left( \frac{x^6}{y^{\frac{1}{5}}} \cdot z^3 \right) }

The Final Answer

After simplifying the given expression using the properties of logarithms, we get:

log⁡7(x6y15⋅z3){ \log_7 \left( \frac{x^6}{y^{\frac{1}{5}}} \cdot z^3 \right) }

This is the simplified form of the given expression, and it is a single logarithm.

Conclusion

In this article, we have simplified a given logarithmic expression into a single logarithm using the properties of logarithms. We have used the product property, quotient property, and power property to rewrite the expression in a more compact and manageable form. This process has helped us to understand the properties of logarithms and how to apply them to simplify complex expressions.

Real-World Applications

Logarithmic expressions have numerous real-world applications in various fields, including physics, engineering, and computer science. For example, logarithmic expressions are used to model population growth, chemical reactions, and electrical circuits. They are also used in data analysis and machine learning to model complex relationships between variables.

Common Mistakes to Avoid

When simplifying logarithmic expressions, there are several common mistakes to avoid:

  • Incorrect application of properties: Make sure to apply the properties of logarithms correctly to avoid errors.
  • Failure to simplify: Don't forget to simplify the expression using the properties of logarithms.
  • Incorrect order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying the expression.

Final Thoughts

Introduction

In our previous article, we explored the concept of simplifying logarithmic expressions into a single logarithm. We reviewed the properties of logarithms and applied them to rewrite a given expression in a more compact and manageable form. In this article, we will continue to delve deeper into the world of logarithmic expressions and answer some frequently asked questions.

Q&A

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

A: A logarithmic expression is a mathematical statement that involves logarithms, whereas a logarithmic function is a function that takes a number as input and returns a logarithm as output.

Q: How do I simplify a logarithmic expression with multiple terms?

A: To simplify a logarithmic expression with multiple terms, you can use the properties of logarithms, such as the product property, quotient property, and power property. For example, if you have the expression log_b(x) + log_b(y), you can use the product property to rewrite it as log_b(xy).

Q: What is the relationship between logarithms and exponents?

A: Logarithms and exponents are inverse operations. This means that if you have an exponential expression, such as 2^x, you can take the logarithm of both sides to get x = log_b(2^x). Similarly, if you have a logarithmic expression, such as log_b(x), you can take the exponential of both sides to get x = b^log_b(x).

Q: How do I evaluate a logarithmic expression with a negative exponent?

A: To evaluate a logarithmic expression with a negative exponent, you can use the property of logarithms that states log_b(x^(-n)) = -n * log_b(x). For example, if you have the expression log_b(x^(-2)), you can rewrite it as -2 * log_b(x).

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

A: A logarithmic expression involves logarithms, whereas an exponential expression involves exponents. For example, the expression log_b(x) is a logarithmic expression, whereas the expression 2^x is an exponential expression.

Q: How do I simplify a logarithmic expression with a fraction?

A: To simplify a logarithmic expression with a fraction, you can use the properties of logarithms, such as the quotient property. For example, if you have the expression log_b(x/y), you can use the quotient property to rewrite it as log_b(x) - log_b(y).

Q: What is the relationship between logarithms and roots?

A: Logarithms and roots are related through the property of logarithms that states log_b(x^(1/n)) = (1/n) * log_b(x). For example, if you have the expression log_b(x^(1/2)), you can rewrite it as (1/2) * log_b(x).

Common Mistakes to Avoid

When working with logarithmic expressions, there are several common mistakes to avoid:

  • Incorrect application of properties: Make sure to apply the properties of logarithms correctly to avoid errors.
  • Failure to simplify: Don't forget to simplify the expression using the properties of logarithms.
  • Incorrect order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying the expression.

Final Thoughts

In conclusion, logarithmic expressions are a fundamental concept in mathematics, and understanding how to simplify them is essential for success in various fields. By reviewing the properties of logarithms and applying them to real-world examples, we can gain a deeper understanding of the underlying mathematics and develop the skills necessary to tackle complex problems.

Real-World Applications

Logarithmic expressions have numerous real-world applications in various fields, including physics, engineering, and computer science. For example, logarithmic expressions are used to model population growth, chemical reactions, and electrical circuits. They are also used in data analysis and machine learning to model complex relationships between variables.

Conclusion

In this article, we have answered some frequently asked questions about logarithmic expressions and provided a comprehensive guide to simplifying them. We have reviewed the properties of logarithms and applied them to real-world examples, demonstrating how to simplify complex expressions and gain a deeper understanding of the underlying mathematics.