Write The Following Sum As A Single Logarithm. Assume All Variables Are Positive.${ \log_3(z) + \log_3(z+4) = \square }$The Answer Format In Lowercase Characters Is: Log_base (number) Spaces In The Answer Are Optional.

Introduction

In mathematics, logarithmic expressions are a fundamental concept in algebra and calculus. The sum of logarithms with the same base can be simplified into a single logarithm, which is a crucial property in solving various mathematical problems. In this article, we will explore how to write the given sum as a single logarithm, assuming all variables are positive.

The Logarithmic Property

The logarithmic property states that the sum of logarithms with the same base can be combined into a single logarithm. This property is expressed as:

$$\log_b(x) + \log_b(y) = \log_b(xy)$$

where $b$ is the base of the logarithm, and $x$ and $y$ are positive real numbers.

Applying the Logarithmic Property

Now, let's apply this property to the given sum:

$$\log_3(z) + \log_3(z+4) = \square$$

Using the logarithmic property, we can combine the two logarithms into a single logarithm:

$$\log_3(z) + \log_3(z+4) = \log_3(z(z+4))$$

Simplifying the Expression

Now, let's simplify the expression inside the logarithm:

$$z(z+4) = z^2 + 4z$$

So, the single logarithm representation of the given sum is:

$$\log_3(z^2 + 4z)$$

Conclusion

In this article, we have successfully simplified the given sum of logarithms into a single logarithm using the logarithmic property. This property is a fundamental concept in mathematics, and understanding it is essential for solving various mathematical problems. By applying this property, we can simplify complex logarithmic expressions and make them easier to work with.

Example Use Cases

The single logarithm representation of the given sum has various applications in mathematics and engineering. For instance, it can be used to solve equations involving logarithms, or to simplify complex expressions in calculus.

Tips and Tricks

When working with logarithmic expressions, it's essential to remember the following tips and tricks:

• Always check if the base of the logarithm is the same for all terms.
• Use the logarithmic property to combine terms with the same base.
• Simplify the expression inside the logarithm to make it easier to work with.

Common Mistakes

When simplifying logarithmic expressions, it's common to make mistakes. Here are some common mistakes to avoid:

• Not checking if the base of the logarithm is the same for all terms.
• Not using the logarithmic property to combine terms with the same base.
• Not simplifying the expression inside the logarithm.

Conclusion

Q: What is the logarithmic property?

A: The logarithmic property states that the sum of logarithms with the same base can be combined into a single logarithm. This property is expressed as:

$$\log_b(x) + \log_b(y) = \log_b(xy)$$

where $b$ is the base of the logarithm, and $x$ and $y$ are positive real numbers.

Q: How do I apply the logarithmic property to simplify a logarithmic expression?

A: To apply the logarithmic property, follow these steps:

1. Check if the base of the logarithm is the same for all terms.
2. Use the logarithmic property to combine terms with the same base.
3. Simplify the expression inside the logarithm.

Q: What is the single logarithm representation of the given sum?

A: The single logarithm representation of the given sum is:

$$\log_3(z^2 + 4z)$$

Q: Can I use the logarithmic property to simplify logarithmic expressions with different bases?

A: No, the logarithmic property only applies to logarithmic expressions with the same base. If the bases are different, you cannot use the logarithmic property to simplify the expression.

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

A: To simplify a logarithmic expression with a negative exponent, use the following property:

$$\log_b(x) = -\log_b(\frac{1}{x})$$

Q: Can I use the logarithmic property to simplify logarithmic expressions with fractional exponents?

A: Yes, you can use the logarithmic property to simplify logarithmic expressions with fractional exponents. However, you need to be careful when simplifying the expression inside the logarithm.

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

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

• Not checking if the base of the logarithm is the same for all terms.
• Not using the logarithmic property to combine terms with the same base.
• Not simplifying the expression inside the logarithm.

Q: How do I check if the base of the logarithm is the same for all terms?

A: To check if the base of the logarithm is the same for all terms, look for the base of the logarithm in each term. If the base is the same, you can use the logarithmic property to simplify the expression.

Q: Can I use the logarithmic property to simplify logarithmic expressions with logarithms of different bases?

A: No, the logarithmic property only applies to logarithmic expressions with the same base. If the bases are different, you cannot use the logarithmic property to simplify the expression.

Q: How do I simplify a logarithmic expression with a logarithm of a product?

A: To simplify a logarithmic expression with a logarithm of a product, use the following property:

$$\log_b(xy) = \log_b(x) + \log_b(y)$$

Q: Can I use the logarithmic property to simplify logarithmic expressions with logarithms of a quotient?

A: Yes, you can use the logarithmic property to simplify logarithmic expressions with logarithms of a quotient. However, you need to be careful when simplifying the expression inside the logarithm.

Conclusion

In conclusion, simplifying logarithmic expressions is a crucial skill in mathematics and engineering. By understanding the logarithmic property and applying it correctly, we can simplify complex expressions and make them easier to work with. Remember to always check if the base of the logarithm is the same for all terms, use the logarithmic property to combine terms with the same base, and simplify the expression inside the logarithm.