Which Expressions Are Equivalent To The Given Expression?Given Expression:${ 5 \log_{10} X + \log_{10} 20 - \log_{10} 10 }$A. { \log_{10}(10x)$}$B. { \log_{10}(2x^5)$}$C. { \log_{10}(2x)^5$}$D.

Understanding Logarithmic Equations

In mathematics, logarithmic equations are a crucial part of algebra and are used to solve various problems. A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In this article, we will explore the given expression and determine which expressions are equivalent to it.

Given Expression

The given expression is:

${ 5 \log_{10} x + \log_{10} 20 - \log_{10} 10 }$

This expression involves logarithms with base 10 and various operations such as addition and subtraction.

Understanding Logarithmic Properties

To simplify the given expression, we need to understand some basic logarithmic properties. These properties include:

• Product Property: $\log_b (xy) = \log_b x + \log_b y$
• Quotient Property: $\log_b \frac{x}{y} = \log_b x - \log_b y$
• Power Property: $\log_b x^y = y \log_b x$

Simplifying the Given Expression

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

${ \log_{10} (x^5) + \log_{10} 20 - \log_{10} 10 }$

Now, we can use the power property to simplify the expression further:

${ \log_{10} (x^5) + \log_{10} \frac{20}{10} }$

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

${ \log_{10} (x^5) + \log_{10} 2 }$

Now, we can use the product property to combine the two logarithms:

${ \log_{10} (x^5 \cdot 2) }$

Simplifying the expression inside the logarithm, we get:

${ \log_{10} (2x^5) }$

Comparing with the Options

Now that we have simplified the given expression, we can compare it with the options:

• Option A: $\log_{10}(10x)$
• Option B: $\log_{10}(2x^5)$
• Option C: $\log_{10}(2x)^5$
• Option D: (Not provided)

Comparing the simplified expression with the options, we can see that:

• Option B: $\log_{10}(2x^5)$ is equivalent to the given expression.

Conclusion

In this article, we simplified the given expression using logarithmic properties and compared it with the options. We found that Option B: $\log_{10}(2x^5)$ is equivalent to the given expression.

Key Takeaways

• Logarithmic properties such as product, quotient, and power properties can be used to simplify logarithmic expressions.
• Understanding these properties is crucial in solving logarithmic equations.
• By simplifying the given expression, we can compare it with the options and determine which one is equivalent.

Final Answer

The final answer is:

• Option B: $\log_{10}(2x^5)$
  Q&A: Logarithmic Equations and Properties =============================================

Understanding Logarithmic Equations

In our previous article, we explored the given expression and determined which expressions are equivalent to it. In this article, we will answer some frequently asked questions related to logarithmic equations and properties.

Q: What is the difference between a logarithmic equation and an exponential equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. An exponential equation, on the other hand, is an equation that involves an exponent. For example, $2^x = 8$ is an exponential equation, while $\log_2 8 = x$ is a logarithmic equation.

Q: What are the basic logarithmic properties?

A: The basic logarithmic properties include:

• Product Property: $\log_b (xy) = \log_b x + \log_b y$
• Quotient Property: $\log_b \frac{x}{y} = \log_b x - \log_b y$
• Power Property: $\log_b x^y = y \log_b x$

Q: How do I simplify a logarithmic expression using the product property?

A: To simplify a logarithmic expression using the product property, you need to rewrite the expression as a product of two or more numbers. For example, $\log_2 (4 \cdot 9)$ can be rewritten as $\log_2 4 + \log_2 9$.

Q: How do I simplify a logarithmic expression using the quotient property?

A: To simplify a logarithmic expression using the quotient property, you need to rewrite the expression as a quotient of two numbers. For example, $\log_2 \frac{12}{4}$ can be rewritten as $\log_2 12 - \log_2 4$.

Q: How do I simplify a logarithmic expression using the power property?

A: To simplify a logarithmic expression using the power property, you need to rewrite the expression as a power of a number. For example, $\log_2 16^3$ can be rewritten as $3 \log_2 16$.

Q: What is the difference between a logarithm with base 10 and a logarithm with base e?

A: A logarithm with base 10 is a logarithm that has a base of 10, while a logarithm with base e is a logarithm that has a base of e (approximately 2.718). The two types of logarithms are related by the following equation: $\log_e x = \frac{\log_{10} x}{\log_{10} e}$.

Q: How do I convert a logarithm with base 10 to a logarithm with base e?

A: To convert a logarithm with base 10 to a logarithm with base e, you can use the following equation: $\log_e x = \frac{\log_{10} x}{\log_{10} e}$.

Q: What is the relationship between logarithms and exponents?

A: Logarithms and exponents are inverse operations. This means that if $a^x = b$, then $\log_a b = x$. Similarly, if $\log_a b = x$, then $a^x = b$.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to isolate the logarithmic term. You can do this by using the properties of logarithms, such as the product property, quotient property, and power property. Once you have isolated the logarithmic term, you can use the definition of a logarithm to solve for the variable.

Conclusion

In this article, we answered some frequently asked questions related to logarithmic equations and properties. We hope that this article has provided you with a better understanding of logarithmic equations and properties.

Key Takeaways

• Logarithmic equations and properties are crucial in solving various problems in mathematics.
• Understanding the basic logarithmic properties, such as the product property, quotient property, and power property, is essential in simplifying logarithmic expressions.
• Converting a logarithm with base 10 to a logarithm with base e can be done using the equation $\log_e x = \frac{\log_{10} x}{\log_{10} e}$.
• Logarithms and exponents are inverse operations, and understanding this relationship is essential in solving logarithmic equations.

Final Answer

The final answer is:

• There is no final numerical answer to this article, as it is a Q&A article that provides information and explanations related to logarithmic equations and properties.