Write The Expression As A Sum Of Multiples Of Logarithms. Assume That Variables Represent Positive Numbers.${ \log_5 X^2(x+3) = \square }$(Type All Variables Without Any Exponents.)

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Introduction

In mathematics, logarithmic expressions are a fundamental concept in algebra and calculus. They are used to represent the power to which a base number must be raised to produce a given value. In this article, we will explore how to express a given logarithmic expression as a sum of multiples of logarithms. We will use the properties of logarithms to simplify the expression and rewrite it in a more manageable form.

The Properties of Logarithms

Before we dive into the problem, let's review the properties of logarithms. The logarithm of a number is the power to which a base number must be raised to produce that number. For example, the logarithm of 100 to the base 10 is 2, because 10^2 = 100. The logarithm of a product is the sum of the logarithms of the individual numbers. This is known as the product rule of logarithms.

The Product Rule of Logarithms

The product rule of logarithms states that:

log(a × b) = log(a) + log(b)

This means that the logarithm of a product is equal to the sum of the logarithms of the individual numbers.

The Power Rule of Logarithms

The power rule of logarithms states that:

log(a^b) = b × log(a)

This means that the logarithm of a number raised to a power is equal to the power times the logarithm of the number.

The Problem

Now that we have reviewed the properties of logarithms, let's tackle the problem at hand. We are given the expression:

log_5 x^2(x+3)

Our goal is to express this expression as a sum of multiples of logarithms.

Step 1: Apply the Power Rule of Logarithms

We can start by applying the power rule of logarithms to the expression. This will allow us to rewrite the expression in a more manageable form.

log_5 x^2(x+3) = log_5 (x^2) + log_5 (x+3)

Step 2: Apply the Product Rule of Logarithms

Now that we have separated the expression into two parts, we can apply the product rule of logarithms to each part. This will allow us to rewrite the expression as a sum of multiples of logarithms.

log_5 (x^2) = 2 × log_5 (x) log_5 (x+3) = log_5 (x) + log_5 (3)

Step 3: Combine the Results

Now that we have applied the power rule and the product rule of logarithms, we can combine the results to get the final expression.

log_5 x^2(x+3) = 2 × log_5 (x) + log_5 (x) + log_5 (3) = 3 × log_5 (x) + log_5 (3)

Conclusion

In this article, we have explored how to express a given logarithmic expression as a sum of multiples of logarithms. We used the properties of logarithms to simplify the expression and rewrite it in a more manageable form. By applying the power rule and the product rule of logarithms, we were able to rewrite the expression as a sum of multiples of logarithms.

Final Answer

The final answer is:

3log⁡5x+log⁡53\boxed{3 \log_5 x + \log_5 3}

Discussion

This problem is a great example of how to apply the properties of logarithms to simplify complex expressions. By using the power rule and the product rule of logarithms, we were able to rewrite the expression in a more manageable form. This is a useful skill to have in mathematics, as it allows us to simplify complex expressions and make them easier to work with.

Related Problems

If you are interested in learning more about logarithmic expressions, here are some related problems that you may find helpful:

  • Express the logarithm of a quotient as a difference of logarithms.
  • Express the logarithm of a product as a sum of logarithms.
  • Express the logarithm of a number raised to a power as a multiple of the logarithm of the number.

Introduction

In our previous article, we explored how to express a given logarithmic expression as a sum of multiples of logarithms. We used the properties of logarithms to simplify the expression and rewrite it in a more manageable form. In this article, we will continue to explore logarithmic expressions and answer some common questions that students often have.

Q&A

Q: What is the difference between a logarithm and an exponent?

A: A logarithm is the power to which a base number must be raised to produce a given value. An exponent, on the other hand, is the power to which a base number is raised to produce a given value.

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, you can use the properties of logarithms, such as the product rule and the power rule. These rules allow you to rewrite the expression in a more manageable form.

Q: What is the product rule of logarithms?

A: The product rule of logarithms states that the logarithm of a product is equal to the sum of the logarithms of the individual numbers. This means that log(a × b) = log(a) + log(b).

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that the logarithm of a number raised to a power is equal to the power times the logarithm of the number. This means that log(a^b) = b × log(a).

Q: How do I express a logarithmic expression as a sum of multiples of logarithms?

A: To express a logarithmic expression as a sum of multiples of logarithms, you can use the properties of logarithms, such as the product rule and the power rule. These rules allow you to rewrite the expression in a more manageable form.

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

A: A logarithmic expression is an expression that involves logarithms, while an exponential expression is an expression that involves exponents. For example, log(x) is a logarithmic expression, while x^2 is an exponential expression.

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you need to find the value of the expression. This can be done by using a calculator or by simplifying the expression using the properties of logarithms.

Q: What is the base of a logarithm?

A: The base of a logarithm is the number that is used as the base of the logarithm. For example, in the expression log(x), the base is 10.

Q: How do I change the base of a logarithm?

A: To change the base of a logarithm, you can use the change of base formula. This formula states that log_b(x) = log_a(x) / log_a(b), where a is the new base.

Conclusion

In this article, we have answered some common questions that students often have about logarithmic expressions. We have also explored the properties of logarithms and how to simplify logarithmic expressions. By understanding these concepts, you will be able to work with logarithmic expressions with confidence.

Final Tips

  • Make sure to use the properties of logarithms to simplify logarithmic expressions.
  • Use a calculator to evaluate logarithmic expressions.
  • Change the base of a logarithm using the change of base formula.
  • Practice, practice, practice! The more you practice working with logarithmic expressions, the more comfortable you will become with them.

Related Resources

If you are interested in learning more about logarithmic expressions, here are some related resources that you may find helpful:

  • Khan Academy: Logarithms
  • Mathway: Logarithmic Expressions
  • Wolfram Alpha: Logarithmic Expressions

I hope this article has been helpful in answering your questions about logarithmic expressions. If you have any further questions or need further clarification, please don't hesitate to ask.