Rephrase Problem: First, let's rephrase the problem into a clear question prompt. question_prompt: What is the simplified form of the natural logarithm of the expression sqrt(x^(2)+1)-x?Identify Expression: Identify the expression inside the natural logarithm function. We have ln(sqrt(x^(2)+1)-x). The expression inside the ln function is sqrt(x^(2)+1)-x.Recognize Complexity: Recognize that the expression inside the ln function is not easily simplifiable. The expression sqrt(x^(2)+1)-x does not have any obvious simplifications that can be applied directly. There are no common factors or terms that can be combined or cancelled out.Consider Logarithmic Properties: Consider properties of logarithms for potential simplification. The properties of logarithms allow us to break down complex expressions under certain conditions, such as when the argument of the logarithm is a product, quotient, or power. However, in this case, the argument sqrt(x^(2)+1)-x is neither a product, quotient, nor a power that can be easily separated into simpler logarithmic terms.Conclude Simplification: Conclude that the expression is already in its simplest form. Since there are no further simplifications or logarithmic properties that can be applied to ln(sqrt(x^(2)+1)-x), we conclude that the expression is already in its simplest form.

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$\ln \left(\sqrt{x^{2}+1}-x\right)$

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Algebra 1

Evaluate integers raised to rational exponents

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\ln \left(\sqrt{x^{2}+1}-x\right)

Rephrase Problem: First, let's rephrase the problem into a clear question prompt. $\newline$ question_prompt: What is the simplified form of the natural logarithm of the expression $\sqrt{x^{2}+1}-x$ ?
Identify Expression: Identify the expression inside the natural logarithm function. $\newline$ We have $\ln(\sqrt{x^{2}+1}-x)$ . The expression inside the $\ln$ function is $\sqrt{x^{2}+1}-x$ .
Recognize Complexity: Recognize that the expression inside the ln function is not easily simplifiable. The expression $\sqrt{x^{2}+1}-x$ does not have any obvious simplifications that can be applied directly. There are no common factors or terms that can be combined or cancelled out.
Consider Logarithmic Properties: Consider properties of logarithms for potential simplification. $\newline$ The properties of logarithms allow us to break down complex expressions under certain conditions, such as when the argument of the logarithm is a product, quotient, or power. However, in this case, the argument $\sqrt{x^{2}+1}-x$ is neither a product, quotient, nor a power that can be easily separated into simpler logarithmic terms.
Conclude Simplification: Conclude that the expression is already in its simplest form. Since there are no further simplifications or logarithmic properties that can be applied to $\ln(\sqrt{x^{2}+1}-x)$ , we conclude that the expression is already in its simplest form.