Rewriting the Logarithmic Expression

Oct 13, 2025 by pinkandtrash.com 68 views

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 focus on rewriting a logarithmic expression as a single logarithm with the same base. We will use the properties of logarithms to simplify the expression and provide a clear understanding of the concept.

Properties of Logarithms

Before we dive into rewriting the logarithmic expression, it's essential to understand the properties of logarithms. The properties of logarithms are as follows:

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

These properties will be used to rewrite the logarithmic expression as a single logarithm with the same base.

Rewriting the Logarithmic Expression

The given logarithmic expression is $\ln 2x + \ln y + 4\ln 7x$ . We will use the properties of logarithms to rewrite this expression as a single logarithm with the same base.

Using the Product Rule, we can rewrite the expression as:

$\ln 2x + \ln y + 4\ln 7x = \ln (2x) + \ln (y) + \ln (7x)^4$

Now, we can use the Product Rule again to combine the logarithms:

$\ln (2x) + \ln (y) + \ln (7x)^4 = \ln (2x \cdot y \cdot (7x)^4)$

Using the Power Rule, we can rewrite the expression as:

$\ln (2x \cdot y \cdot (7x)^4) = \ln (2x \cdot y \cdot 7^4 \cdot x^4)$

Now, we can use the Product Rule again to combine the logarithms:

$\ln (2x \cdot y \cdot 7^4 \cdot x^4) = \ln (2 \cdot 7^4 \cdot x^5 \cdot y)$

Simplifying the Expression

The expression $\ln (2 \cdot 7^4 \cdot x^5 \cdot y)$ can be simplified by combining the constants and variables.

Using the Product Rule, we can rewrite the expression as:

$\ln (2 \cdot 7^4 \cdot x^5 \cdot y) = \ln (2 \cdot 7^4) + \ln (x^5) + \ln (y)$

Now, we can use the Power Rule to rewrite the expression as:

$\ln (2 \cdot 7^4) + \ln (x^5) + \ln (y) = \ln (2) + 4\ln (7) + 5\ln (x) + \ln (y)$

Conclusion

In this article, we have rewritten the logarithmic expression $\ln 2x + \ln y + 4\ln 7x$ as a single logarithm with the same base. We have used the properties of logarithms to simplify the expression and provide a clear understanding of the concept.

The rewritten expression is $\ln (2 \cdot 7^4 \cdot x^5 \cdot y) = \ln (2) + 4\ln (7) + 5\ln (x) + \ln (y)$ . This expression can be further simplified by combining the constants and variables.

Final Answer

Introduction

In our previous article, we discussed how to rewrite a logarithmic expression as a single logarithm with the same base. We used the properties of logarithms to simplify the expression and provide a clear understanding of the concept. In this article, we will answer some frequently asked questions related to rewriting logarithmic expressions.

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

A: A logarithmic expression is a combination of logarithms with different bases or arguments, while a single logarithm is a logarithm with a single base and argument.

Q: How do I know which property of logarithms to use when rewriting an expression?

A: To determine which property of logarithms to use, you need to analyze the expression and identify the operations involved. If the expression involves multiplication, use the Product Rule. If the expression involves division, use the Quotient Rule. If the expression involves exponentiation, use the Power Rule.

Q: Can I rewrite a logarithmic expression with different bases as a single logarithm?

A: Yes, you can rewrite a logarithmic expression with different bases as a single logarithm using the Change of Base Formula. The Change of Base Formula states that $\log_b x = \frac{\log_c x}{\log_c b}$ , where $c$ is any positive real number not equal to 1.

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

A: To simplify a logarithmic expression with fractions, you need to use the Quotient Rule. The Quotient Rule states that $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$ . You can also use the Product Rule to rewrite the expression as a product of logarithms.

Q: Can I rewrite a logarithmic expression with negative exponents as a single logarithm?

A: Yes, you can rewrite a logarithmic expression with negative exponents as a single logarithm using the Power Rule. The Power Rule states that $\log_b (x^y) = y \log_b x$ . You can also use the Change of Base Formula to rewrite the expression with a positive exponent.

Q: How do I evaluate a logarithmic expression with multiple logarithms?

A: To evaluate a logarithmic expression with multiple logarithms, you need to use the properties of logarithms to simplify the expression. You can use the Product Rule, Quotient Rule, and Power Rule to rewrite the expression as a single logarithm.

Q: Can I use a calculator to evaluate a logarithmic expression?

A: Yes, you can use a calculator to evaluate a logarithmic expression. However, you need to make sure that the calculator is set to the correct base and that the expression is entered correctly.

Conclusion

In this article, we have answered some frequently asked questions related to rewriting logarithmic expressions. We have discussed the properties of logarithms, how to simplify expressions with fractions, and how to evaluate expressions with multiple logarithms. We hope that this article has provided a clear understanding of the concept and has helped you to become more confident in your ability to rewrite logarithmic expressions.

Final Answer

The final answer is $\boxed{yes}$ , you can rewrite a logarithmic expression as a single logarithm with the same base using the properties of logarithms.