Which Expression Is Equivalent To The Expression Below?${ L G+1 G+1 G+1 G+\mid G+1 G }$A. ${ 6+g }$B. ${ G^6 }$C. ${ 6g }$D. ${ \frac{g}{6} }$

Mar 10, 2025 by ADMIN 148 views

**Simplifying Logarithmic Expressions: A Step-by-Step Guide**

Introduction

Logarithmic expressions can be complex and challenging to simplify. However, with a clear understanding of the properties of logarithms, we can break down these expressions into more manageable parts. In this article, we will explore how to simplify a given logarithmic expression and identify the equivalent expression from a set of options.

Understanding Logarithmic Properties

Before we dive into the problem, let's review some essential properties of logarithms:

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

These properties will be crucial in simplifying the given expression.

The Given Expression

The expression we need to simplify is:

$\log_{10} (g+1) + \log_{10} (g+1) + \log_{10} (g+1) + |\log_{10} (g+1)|$

Step 1: Simplify the Expression Using the Product Property

We can start by combining the first three logarithmic terms using the product property:

$\log_{10} (g+1) + \log_{10} (g+1) + \log_{10} (g+1) = \log_{10} ((g+1)(g+1)(g+1))$

This simplifies to:

$\log_{10} ((g+1)^3)$

Step 2: Simplify the Absolute Value Expression

The absolute value expression $|\log_{10} (g+1)|$ can be simplified as follows:

$|\log_{10} (g+1)| = \begin{cases} \log_{10} (g+1) & \text{if } g+1 \geq 1 \\ -\log_{10} (g+1) & \text{if } g+1 < 1 \end{cases}$

However, since $g+1$ is always greater than or equal to 1, we can simplify the absolute value expression to:

$|\log_{10} (g+1)| = \log_{10} (g+1)$

Step 3: Combine the Simplified Expressions

Now that we have simplified the individual expressions, we can combine them:

$\log_{10} ((g+1)^3) + \log_{10} (g+1) = \log_{10} ((g+1)^3(g+1))$

This simplifies to:

$\log_{10} ((g+1)^4)$

Conclusion

The simplified expression is $\log_{10} ((g+1)^4)$ . Now, let's compare this expression with the given options:

A. ${ 6+g }$ B. ${ g^6 }$ C. ${ 6g }$ D. ${ \frac{g}{6} }$

Which Expression is Equivalent to the Simplified Expression?

To determine which expression is equivalent to the simplified expression, we need to rewrite the simplified expression in a form that matches one of the given options.

Using the power property of logarithms, we can rewrite the simplified expression as:

$\log_{10} ((g+1)^4) = 4 \log_{10} (g+1)$

However, this expression does not match any of the given options. We need to rewrite the expression in a different form.

Using the product property of logarithms, we can rewrite the simplified expression as:

$\log_{10} ((g+1)^4) = \log_{10} (g+1) + \log_{10} (g+1) + \log_{10} (g+1) + \log_{10} (g+1)$

This expression matches option A: ${ 6+g }$. However, this is not the correct answer.

We need to rewrite the expression in a form that matches one of the given options.

Using the power property of logarithms, we can rewrite the simplified expression as:

$\log_{10} ((g+1)^4) = \log_{10} (g^4 + 4g^3 + 6g^2 + 4g + 1)$

This expression does not match any of the given options.

However, we can rewrite the expression in a different form:

Q: What is the product property of logarithms?

A: The product property of logarithms states that $\log_b (xy) = \log_b x + \log_b y$ . This means that the logarithm of a product can be rewritten as the sum of the logarithms of the individual factors.

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 identify the factors inside the logarithm and rewrite the expression as the sum of the logarithms of the individual factors.

Q: What is the quotient property of logarithms?

A: The quotient property of logarithms states that $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$ . This means that the logarithm of a quotient can be rewritten as the difference of the logarithms of the individual factors.

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 identify the factors inside the logarithm and rewrite the expression as the difference of the logarithms of the individual factors.

Q: What is the power property of logarithms?

A: The power property of logarithms states that $\log_b (x^y) = y \log_b x$ . This means that the logarithm of a power can be rewritten as the product of the exponent and the logarithm of the base.

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 identify the exponent and the base inside the logarithm and rewrite the expression as the product of the exponent and the logarithm of the base.

Q: Can I simplify a logarithmic expression with multiple logarithms?

A: Yes, you can simplify a logarithmic expression with multiple logarithms by using the product, quotient, and power properties of logarithms.

Q: How do I determine which property to use when simplifying a logarithmic expression?

A: To determine which property to use when simplifying a logarithmic expression, you need to analyze the expression and identify the factors, exponents, and bases inside the logarithm. Then, you can choose the appropriate property to simplify the expression.

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

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

Not using the correct property of logarithms
Not simplifying the expression correctly
Not checking the domain of the logarithmic function
Not considering the restrictions on the variables

Q: How can I practice simplifying logarithmic expressions?

A: You can practice simplifying logarithmic expressions by working through examples and exercises in a textbook or online resource. You can also try simplifying logarithmic expressions on your own and checking your work with a calculator or online tool.

Conclusion

Simplifying logarithmic expressions can be a challenging task, but with practice and patience, you can become proficient in using the product, quotient, and power properties of logarithms. Remember to analyze the expression carefully, choose the correct property, and simplify the expression correctly. With these skills, you can tackle even the most complex logarithmic expressions with confidence.