Which Of The Following Expressions Is Equivalent To \ln \left(10 X^2\right ] For All Values Of X X X , Including X \textless 0 X \ \textless \ 0 X \textless 0 ? Select All That Apply.- Ln ⁡ ( 5 X ) + Ln ⁡ ( 2 X \ln (5 X) + \ln (2 X Ln ( 5 X ) + Ln ( 2 X ]- $\ln \left(20 X^3\right) -

Which of the Following Expressions is Equivalent to $\ln \left(10 x^2\right)$ for All Values of $x$, Including $x < 0$?

In mathematics, logarithmic functions play a crucial role in various mathematical operations, including differentiation and integration. The natural logarithm, denoted by $\ln$, is a fundamental function that is used extensively in mathematical modeling and problem-solving. In this article, we will explore the equivalent expressions of $\ln \left(10 x^2\right)$ for all values of $x$, including $x < 0$.

Properties of Logarithmic Functions

Before we dive into the equivalent expressions, it's essential to understand the properties of logarithmic functions. The logarithmic function $\ln$ has several properties that are crucial in solving logarithmic equations and inequalities. Some of the key properties of logarithmic functions include:

• Product Rule: $\ln (ab) = \ln a + \ln b$
• Quotient Rule: $\ln \left(\frac{a}{b}\right) = \ln a - \ln b$
• Power Rule: $\ln (a^b) = b \ln a$

Equivalent Expressions

Now, let's explore the equivalent expressions of $\ln \left(10 x^2\right)$ for all values of $x$, including $x < 0$. We will use the properties of logarithmic functions to simplify the expressions.

$\ln (5 x) + \ln (2 x)$

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

$\ln (5 x) + \ln (2 x) = \ln (5 x \cdot 2 x) = \ln (10 x^2)$

This expression is equivalent to $\ln \left(10 x^2\right)$ for all values of $x$, including $x < 0$.

$\ln \left(20 x^3\right)$

Using the power rule, we can rewrite the expression as:

$\ln \left(20 x^3\right) = \ln (20) + \ln (x^3) = \ln (20) + 3 \ln x$

This expression is not equivalent to $\ln \left(10 x^2\right)$ for all values of $x$, including $x < 0$. The reason is that the expression $\ln (20) + 3 \ln x$ is not equal to $\ln (10 x^2)$ for all values of $x$, especially when $x < 0$.

In conclusion, the expression $\ln (5 x) + \ln (2 x)$ is equivalent to $\ln \left(10 x^2\right)$ for all values of $x$, including $x < 0$. This is because the product rule allows us to simplify the expression to $\ln (10 x^2)$, which is equal to the original expression.

On the other hand, the expression $\ln \left(20 x^3\right)$ is not equivalent to $\ln \left(10 x^2\right)$ for all values of $x$, including $x < 0$. This is because the power rule does not allow us to simplify the expression to $\ln (10 x^2)$, especially when $x < 0$.

The final answer is:

• $\ln (5 x) + \ln (2 x)$
• $\ln \left(20 x^3\right)$ (Note: This expression is not equivalent to $\ln \left(10 x^2\right)$ for all values of $x$, including $x < 0$.)

The discussion category for this article is mathematics. The article explores the equivalent expressions of $\ln \left(10 x^2\right)$ for all values of $x$, including $x < 0$. The article uses the properties of logarithmic functions to simplify the expressions and arrive at the final answer.

• [1] "Logarithmic Functions" by Math Open Reference
• [2] "Properties of Logarithmic Functions" by Wolfram MathWorld
• [3] "Equivalent Expressions" by Khan Academy
  Q&A: Equivalent Expressions of $\ln \left(10 x^2\right)$ ===========================================================

In our previous article, we explored the equivalent expressions of $\ln \left(10 x^2\right)$ for all values of $x$, including $x < 0$. In this article, we will answer some frequently asked questions (FAQs) related to the equivalent expressions of $\ln \left(10 x^2\right)$.

Q: What is the product rule of logarithmic functions?

A: The product rule of logarithmic functions states that $\ln (ab) = \ln a + \ln b$. This rule allows us to simplify the expression $\ln (5 x) + \ln (2 x)$ to $\ln (10 x^2)$.

Q: How do I apply the product rule to simplify the expression $\ln (5 x) + \ln (2 x)$?

A: To apply the product rule, simply multiply the two expressions inside the logarithms: $\ln (5 x) + \ln (2 x) = \ln (5 x \cdot 2 x) = \ln (10 x^2)$.

Q: What is the power rule of logarithmic functions?

A: The power rule of logarithmic functions states that $\ln (a^b) = b \ln a$. This rule allows us to simplify the expression $\ln \left(20 x^3\right)$ to $\ln (20) + 3 \ln x$.

Q: How do I apply the power rule to simplify the expression $\ln \left(20 x^3\right)$?

A: To apply the power rule, simply rewrite the expression inside the logarithm as a product of the base and the exponent: $\ln \left(20 x^3\right) = \ln (20) + \ln (x^3) = \ln (20) + 3 \ln x$.

Q: Why is the expression $\ln \left(20 x^3\right)$ not equivalent to $\ln \left(10 x^2\right)$ for all values of $x$, including $x < 0$?

A: The expression $\ln \left(20 x^3\right)$ is not equivalent to $\ln \left(10 x^2\right)$ for all values of $x$, including $x < 0$, because the power rule does not allow us to simplify the expression to $\ln (10 x^2)$, especially when $x < 0$.

Q: What are some common mistakes to avoid when working with logarithmic functions?

A: Some common mistakes to avoid when working with logarithmic functions include:

• Forgetting to apply the product rule when simplifying expressions with multiple logarithms
• Forgetting to apply the power rule when simplifying expressions with exponents inside the logarithm
• Not checking the domain of the logarithmic function to ensure that it is defined for all values of $x$

In conclusion, the equivalent expressions of $\ln \left(10 x^2\right)$ for all values of $x$, including $x < 0$, are $\ln (5 x) + \ln (2 x)$ and $\ln \left(20 x^3\right)$ is not equivalent. We hope that this Q&A article has helped to clarify any questions or doubts you may have had about the equivalent expressions of $\ln \left(10 x^2\right)$.

The final answer is:

• $\ln (5 x) + \ln (2 x)$
• $\ln \left(20 x^3\right)$ (Note: This expression is not equivalent to $\ln \left(10 x^2\right)$ for all values of $x$, including $x < 0$.)

The discussion category for this article is mathematics. The article answers some frequently asked questions (FAQs) related to the equivalent expressions of $\ln \left(10 x^2\right)$.

• [1] "Logarithmic Functions" by Math Open Reference
• [2] "Properties of Logarithmic Functions" by Wolfram MathWorld
• [3] "Equivalent Expressions" by Khan Academy