Expand The Following Logarithmic Expression Into A Sum Or Difference Of Logs: \log_7\left(\frac{y}{z}\right ]A. Log ⁡ 7 ( Y ) + Log ⁡ 7 ( Z \log_7(y) + \log_7(z Lo G 7 ( Y ) + Lo G 7 ( Z ]B. Log ⁡ 7 ( Y ) Log ⁡ 7 ( Z ) \frac{\log_7(y)}{\log_7(z)} L O G 7 ( Z ) L O G 7 ( Y ) C. Log ⁡ 7 ( Y ) − Log ⁡ 7 ( Z \log_7(y) - \log_7(z Lo G 7 ( Y ) − Lo G 7 ( Z ]D. $\log_7(y)

Apr 2, 2025 by ADMIN 382 views

Logarithmic Expressions: Expanding and Simplifying

Logarithmic expressions are a fundamental concept in mathematics, and understanding how to expand and simplify them is crucial for solving various mathematical problems. In this article, we will focus on expanding the logarithmic expression $\log_7\left(\frac{y}{z}\right)$ into a sum or difference of logs.

The Logarithmic Expression

The given logarithmic expression is $\log_7\left(\frac{y}{z}\right)$ . This expression represents the logarithm of the ratio of $y$ to $z$ with base $7$ . To expand this expression, we need to apply the properties of logarithms.

Applying the Properties of Logarithms

One of the fundamental properties of logarithms is the product rule, which states that $\log_b(xy) = \log_b(x) + \log_b(y)$ . However, in this case, we have a ratio, not a product. To apply the product rule, we need to rewrite the ratio as a product.

Rewriting the Ratio as a Product

We can rewrite the ratio $\frac{y}{z}$ as $y \cdot \frac{1}{z}$ . Now, we can apply the product rule to expand the logarithmic expression.

Expanding the Logarithmic Expression

Using the product rule, we can expand the logarithmic expression as follows:

\log_7\left(\frac{y}{z}\right) = \log_7(y \cdot \frac{1}{z})

Applying the product rule, we get:

\log_7(y \cdot \frac{1}{z}) = \log_7(y) + \log_7(\frac{1}{z})

Simplifying the Expression

Now, we can simplify the expression by applying the property of logarithms that states $\log_b(\frac{1}{x}) = -\log_b(x)$ . In this case, we have:

\log_7(\frac{1}{z}) = -\log_7(z)

Substituting this expression into the previous equation, we get:

\log_7(y) + \log_7(\frac{1}{z}) = \log_7(y) - \log_7(z)

In conclusion, the logarithmic expression $\log_7\left(\frac{y}{z}\right)$ can be expanded into a sum or difference of logs as $\log_7(y) - \log_7(z)$ . This result is based on the properties of logarithms, specifically the product rule and the property of logarithms that states $\log_b(\frac{1}{x}) = -\log_b(x)$ .

The correct answer is C. $\log_7(y) - \log_7(z)$ .

This problem is a great example of how to apply the properties of logarithms to expand and simplify logarithmic expressions. It requires a good understanding of the product rule and the property of logarithms that states $\log_b(\frac{1}{x}) = -\log_b(x)$ . By applying these properties, we can expand and simplify logarithmic expressions and solve various mathematical problems.

*arithmic expressions

Properties of logarithms
Product rule
Logarithm of a ratio
Simplifying logarithmic expressions

Expand the logarithmic expression $\log_5\left(\frac{x}{y}\right)$ into a sum or difference of logs.
Simplify the logarithmic expression $\log_3(12x)$ using the properties of logarithms.
Expand the logarithmic expression $\log_2\left(\frac{a}{b}\right)$ into a sum or difference of logs.

Expand the logarithmic expression $\log_4\left(\frac{m}{n}\right)$ into a sum or difference of logs.
Simplify the logarithmic expression $\log_6(24x)$ using the properties of logarithms.
Expand the logarithmic expression $\log_3\left(\frac{p}{q}\right)$ into a sum or difference of logs.

In conclusion, expanding and simplifying logarithmic expressions is a crucial skill in mathematics. By applying the properties of logarithms, we can expand and simplify logarithmic expressions and solve various mathematical problems. This article has provided a step-by-step guide on how to expand the logarithmic expression $\log_7\left(\frac{y}{z}\right)$ into a sum or difference of logs. We hope that this article has been helpful in understanding the properties of logarithms and how to apply them to expand and simplify logarithmic expressions.
Logarithmic Expressions: Q&A

In our previous article, we discussed how to expand and simplify logarithmic expressions using the properties of logarithms. In this article, we will provide a Q&A section to help you better understand the concepts and apply them to various problems.

Q1: What is the product rule of logarithms?

A1: The product rule of logarithms states that $\log_b(xy) = \log_b(x) + \log_b(y)$ . This means that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Q2: How do I apply the product rule to expand a logarithmic expression?

A2: To apply the product rule, you need to rewrite the logarithmic expression as a product. For example, if you have $\log_b\left(\frac{x}{y}\right)$ , you can rewrite it as $\log_b(x \cdot \frac{1}{y})$ . Then, you can apply the product rule to expand the expression.

Q3: What is the property of logarithms that states $\log_b(\frac{1}{x}) = -\log_b(x)$ ?

A3: This property states that the logarithm of a reciprocal is equal to the negative of the logarithm of the original value. In other words, if you have $\log_b(\frac{1}{x})$ , you can rewrite it as $-\log_b(x)$ .

Q4: How do I simplify a logarithmic expression using the properties of logarithms?

A4: To simplify a logarithmic expression, you need to apply the properties of logarithms, such as the product rule and the property of logarithms that states $\log_b(\frac{1}{x}) = -\log_b(x)$ . You can also use the power rule, which states that $\log_b(x^n) = n\log_b(x)$ .

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 $\log_b(x)$ . An exponential expression, on the other hand, is an expression that involves an exponent, such as $b^x$ . While logarithmic and exponential expressions are related, they are not the same thing.

Q6: How do I evaluate a logarithmic expression?

A6: To evaluate a logarithmic expression, you need to find the value of the logarithm. This can be done using a calculator or by applying the properties of logarithms. For example, if you have $\log_b(x)$ , you can evaluate it by finding the value of $x$ that satisfies the equation $b^x = x$ .

Q7: What is the relationship between logarithmic and exponential expressions?

A7: Logarithmic and exponential expressions are related through the property of logarithms that states $\log_b(b^x) = x$ . This means that the logarithm of an exponential expression is equal to the exponent. For example, if you have $\log_b(b^x)$ , you can rewrite it as $x$ .

Q8: How do I use logarithmic expressions in real-world problems

A8: Logarithmic expressions are used in a wide range of real-world problems, including finance, science, and engineering. For example, logarithmic expressions can be used to model population growth, chemical reactions, and financial transactions.

In conclusion, logarithmic expressions are a powerful tool for solving mathematical problems. By understanding the properties of logarithms and how to apply them, you can expand and simplify logarithmic expressions and solve a wide range of problems. We hope that this Q&A article has been helpful in understanding the concepts and applying them to various problems.