ln sqrt((x^(2)+1)/(x^(3)+5))

Use Logarithmic Properties: We start by expressing the given expression using logarithmic properties. The natural logarithm of a square root can be expressed as one-half the natural logarithm of the expression inside the square root. ln(sqrt((x^2 + 1)/(x^3 + 5))) = (1/2) * ln((x^2 + 1)/(x^3 + 5))Apply Quotient Property: Next, we use the property of logarithms that allows us to write the logarithm of a quotient as the difference of logarithms. (1/2) * ln((x^2 + 1)/(x^3 + 5)) = (1/2) * (ln(x^2 + 1) - ln(x^3 + 5))Final Simplified Form: Now we have the expression in a simplified logarithmic form. There are no further simplifications that can be made without knowing the value of x, so this is the final simplified form of the expression.

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$\ln\sqrt{\frac{x^{2}+1}{x^{3}+5}}$

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Precalculus

Evaluate rational exponents

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\ln\sqrt{\frac{x^{2}+1}{x^{3}+5}}

Use Logarithmic Properties: We start by expressing the given expression using logarithmic properties. $\newline$ The natural logarithm of a square root can be expressed as one-half the natural logarithm of the expression inside the square root. $\newline$ $\ln(\sqrt{(x^2 + 1)/(x^3 + 5)}) = (\frac{1}{2}) \cdot \ln((x^2 + 1)/(x^3 + 5))$
Apply Quotient Property: Next, we use the property of logarithms that allows us to write the logarithm of a quotient as the difference of logarithms. $\newline$ $(\frac{1}{2}) \cdot \ln(\frac{x^2 + 1}{x^3 + 5}) = (\frac{1}{2}) \cdot (\ln(x^2 + 1) - \ln(x^3 + 5))$
Final Simplified Form: Now we have the expression in a simplified logarithmic form. There are no further simplifications that can be made without knowing the value of $x$ , so this is the final simplified form of the expression.