Simplify The Expression:$\[ 2 \log_3 5 - \log_3 10 + 3 \log_3 4 \\]

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Introduction

In mathematics, logarithms are a fundamental concept that plays a crucial role in various mathematical operations. The expression given in this problem involves logarithms with different bases and operations. Our goal is to simplify the given expression using the properties of logarithms.

Understanding Logarithms

Before we dive into simplifying the expression, let's briefly review the concept of logarithms. A logarithm is the inverse operation of exponentiation. In other words, if we have a number x and a base b, the logarithm of x with base b is the exponent to which b must be raised to produce x. This is denoted as logb x.

For example, if we have 2^3 = 8, then log2 8 = 3. Similarly, if we have 10^2 = 100, then log10 100 = 2.

Properties of Logarithms

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

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

Simplifying the Expression

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

2 log3 5 - log3 10 + 3 log3 4

Using the product property, we can rewrite the expression as:

log3 (5^2) - log3 10 + log3 (4^3)

Now, using the power property, we can rewrite the expression as:

2 log3 5 - log3 10 + 3 log3 4

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

log3 (5^2 / 10) + log3 (4^3)

Now, using the product property, we can rewrite the expression as:

log3 (25 / 10) + log3 (64)

Simplifying further, we get:

log3 2.5 + log3 64

Using the product property, we can rewrite the expression as:

log3 (2.5 * 64)

Now, using the product property, we can rewrite the expression as:

log3 160

Conclusion

In this article, we simplified the given expression using the properties of logarithms. We reviewed the concept of logarithms and the properties of logarithms, and then applied these properties to simplify the expression. The final simplified expression is log3 160.

Frequently Asked Questions

  • What is the concept of logarithms?
  • What are the properties of logarithms?
  • How do we simplify the given expression using the properties of logarithms?

Final Answer

The final answer is log3 160.

Introduction

In our previous article, we simplified the given expression using the properties of logarithms. We reviewed the concept of logarithms and the properties of logarithms, and then applied these properties to simplify the expression. In this article, we will answer some frequently asked questions related to the simplification of the expression.

Q&A

Q1: What is the concept of logarithms?

A1: A logarithm is the inverse operation of exponentiation. In other words, if we have a number x and a base b, the logarithm of x with base b is the exponent to which b must be raised to produce x. This is denoted as logb x.

Q2: What are the properties of logarithms?

A2: There are several properties of logarithms that we use to simplify the given expression. These properties are:

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

Q3: How do we simplify the given expression using the properties of logarithms?

A3: To simplify the given expression, we first use the product property to rewrite the expression as log3 (5^2) - log3 10 + log3 (4^3). Then, we use the power property to rewrite the expression as 2 log3 5 - log3 10 + 3 log3 4. Next, we use the quotient property to rewrite the expression as log3 (5^2 / 10) + log3 (4^3). Finally, we use the product property to rewrite the expression as log3 (25 / 10) + log3 (64), and then simplify further to get log3 2.5 + log3 64.

Q4: What is the final simplified expression?

A4: The final simplified expression is log3 160.

Q5: How do we apply the properties of logarithms to simplify the expression?

A5: To apply the properties of logarithms, we first identify the operations involved in the expression. Then, we use the appropriate property to rewrite the expression. For example, if we have an expression with a product, we use the product property to rewrite it as a sum of logarithms.

Q6: What are some common mistakes to avoid when simplifying logarithmic expressions?

A6: Some common mistakes to avoid when simplifying logarithmic expressions include:

  • Not using the correct property of logarithms
  • Not following the order of operations
  • Not simplifying the expression correctly

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the given expression. We reviewed the concept of logarithms and the properties of logarithms, and then applied these properties to simplify the expression. We also discussed some common mistakes to avoid when simplifying logarithmic expressions.

Final Answer

The final answer is log3 160.

Additional Resources

Final Tips

  • When simplifying logarithmic expressions, make sure to use the correct property of logarithms.
  • Follow the order of operations when simplifying logarithmic expressions.
  • Simplify the expression correctly to avoid common mistakes.