Simplify The Expression $5 \ln Z + \frac{\ln X}{3}$.A) $\ln \left(z^5 \sqrt[3]{x}\right)$ B) \$\ln \left(y^3 X^{15}\right)$[/tex\] C) $\ln \frac{x^5}{y^3}$ D) $\ln \left(y^{15} X^3\right)$

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Introduction

Logarithmic expressions can be complex and challenging to simplify. However, with a clear understanding of the properties of logarithms, we can simplify even the most daunting expressions. In this article, we will focus on simplifying the expression $5 \ln z + \frac{\ln x}{3}$. We will explore the properties of logarithms, apply them to the given expression, and arrive at the simplified form.

Understanding Logarithmic Properties

Before we dive into simplifying the expression, it's essential to understand the properties of logarithms. The two primary properties of logarithms are:

  • Product Property: log⁡b(xy)=log⁡bx+log⁡by\log_b (xy) = \log_b x + \log_b y
  • Power Property: log⁡b(xy)=ylog⁡bx\log_b (x^y) = y \log_b x

These properties will be instrumental in simplifying the given expression.

Simplifying the Expression

Now that we have a solid understanding of logarithmic properties, let's apply them to the given expression $5 \ln z + \frac{\ln x}{3}$.

Using the Power Property, we can rewrite the first term as:

5ln⁡z=ln⁡z55 \ln z = \ln z^5

Next, we can rewrite the second term using the Power Property:

ln⁡x3=ln⁡x13\frac{\ln x}{3} = \ln x^{\frac{1}{3}}

Now, we can combine the two terms using the Product Property:

ln⁡z5+ln⁡x13=ln⁡(z5x13)\ln z^5 + \ln x^{\frac{1}{3}} = \ln (z^5 x^{\frac{1}{3}})

However, we can further simplify the expression by combining the exponents of xx:

ln⁡(z5x13)=ln⁡(z5x3)\ln (z^5 x^{\frac{1}{3}}) = \ln (z^5 \sqrt[3]{x})

Conclusion

In conclusion, we have successfully simplified the expression $5 \ln z + \frac{\ln x}{3}$ using the properties of logarithms. The simplified form is $\ln (z^5 \sqrt[3]{x})$.

Comparison with Answer Choices

Now that we have the simplified expression, let's compare it with the answer choices:

  • A) $\ln \left(z^5 \sqrt[3]{x}\right)$
  • B) $\ln \left(y^3 x^{15}\right)$
  • C) $\ln \frac{x5}{y3}$
  • D) $\ln \left(y^{15} x^3\right)$

The only answer choice that matches our simplified expression is:

  • A) $\ln \left(z^5 \sqrt[3]{x}\right)$

Therefore, the correct answer is A) $\ln \left(z^5 \sqrt[3]{x}\right)$.

Final Thoughts

Simplifying logarithmic expressions can be a daunting task, but with a clear understanding of the properties of logarithms, we can simplify even the most complex expressions. In this article, we applied the properties of logarithms to simplify the expression $5 \ln z + \frac{\ln x}{3}$. We arrived at the simplified form $\ln (z^5 \sqrt[3]{x})$ and compared it with the answer choices to determine the correct answer.

Additional Resources

For further practice and review, we recommend exploring the following resources:

  • Khan Academy: Logarithms
  • Mathway: Logarithmic Properties
  • Wolfram Alpha: Logarithmic Simplification

By practicing and reviewing these resources, you will become more confident in simplifying logarithmic expressions and tackling complex mathematical problems.

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Introduction

In our previous article, we explored the properties of logarithms and applied them to simplify the expression $5 \ln z + \frac{\ln x}{3}$. We arrived at the simplified form $\ln (z^5 \sqrt[3]{x})$ and compared it with the answer choices to determine the correct answer.

In this article, we will continue to explore logarithmic simplification by answering some of the most frequently asked questions. Whether you're a student, teacher, or simply looking to brush up on your math skills, this Q&A guide is for you.

Q&A

Q1: What is the product property of logarithms?

A1: The product property of logarithms states that log⁡b(xy)=log⁡bx+log⁡by\log_b (xy) = \log_b x + \log_b y. This means that the logarithm of a product is equal to the sum of the logarithms of the individual terms.

Q2: How do I simplify a logarithmic expression with multiple terms?

A2: To simplify a logarithmic expression with multiple terms, you can use the product property to combine the terms. For example, if you have the expression log⁡b(x2y3)\log_b (x^2 y^3), you can rewrite it as log⁡bx2+log⁡by3\log_b x^2 + \log_b y^3.

Q3: What is the power property of logarithms?

A3: The power property of logarithms states that log⁡b(xy)=ylog⁡bx\log_b (x^y) = y \log_b x. This means that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base.

Q4: How do I simplify a logarithmic expression with a power?

A4: To simplify a logarithmic expression with a power, you can use the power property to rewrite the expression. For example, if you have the expression log⁡b(x3)\log_b (x^3), you can rewrite it as 3log⁡bx3 \log_b x.

Q5: What is the difference between a logarithmic expression and an exponential expression?

A5: A logarithmic expression is an expression that involves a logarithm, such as log⁡bx\log_b x. An exponential expression, on the other hand, is an expression that involves an exponent, such as xyx^y.

Q6: How do I simplify a logarithmic expression with a fraction?

A6: To simplify a logarithmic expression with a fraction, you can use the product property to rewrite the expression. For example, if you have the expression log⁡b(xy)\log_b \left(\frac{x}{y}\right), you can rewrite it as log⁡bx−log⁡by\log_b x - \log_b y.

Q7: What is the relationship between logarithms and exponents?

A7: Logarithms and exponents are inverse operations. This means that if you have an exponential expression, you can take the logarithm of it to get back to the original value. Conversely, if you have a logarithmic expression, you can take the exponent of it to get back to the original value.

Conclusion

In conclusion, logarithmic simplification is a powerful tool that can be used to simplify complex expressions. By understanding the properties of logarithms and applying them to real-world problems, you can become more confident in your math skills and tackle even the most challenging problems.

Additional Resources

For further practice and review, we recommend exploring the following resources:

  • Khan Academy: Logarithms
  • Mathway: Logarithmic Properties
  • Wolfram Alpha: Logarithmic Simplification

By practicing and reviewing these resources, you will become more confident in simplifying logarithmic expressions and tackling complex mathematical problems.

Final Thoughts

Logarithmic simplification is a fascinating topic that can be applied to a wide range of real-world problems. Whether you're a student, teacher, or simply looking to brush up on your math skills, we hope this Q&A guide has been helpful in answering your questions and providing you with a deeper understanding of logarithmic simplification.