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Expand
ln((sqrt((x-1)(2x+1)^(2)))/((x^(2)-9)))

Expand $\ln \left(\frac{\sqrt{(x-1)(2 x+1)^{2}}}{\left(x^{2}-9\right)}\right)$

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Add and subtract three or more integers

Full solution

Q. Expand

\ln \left(\frac{\sqrt{(x-1)(2 x+1)^{2}}}{\left(x^{2}-9\right)}\right)

Apply Logarithmic Property: We need to expand the logarithmic expression $\ln\left(\frac{\sqrt{(x-1)(2x+1)^{2}}}{(x^{2}-9)}\right)$ . To do this, we will use properties of logarithms and simplify the expression step by step.
Separate Logarithms: First, we apply the property of logarithms that says $\ln(\frac{a}{b}) = \ln(a) - \ln(b)$ . This gives us two separate logarithms to work with. $\newline$ \ln\left(\frac{\sqrt{(x\(-1\))(\(2\)x+\(1\))^{\(2\)}}}{x^{\(2\)}\(-9\)}\right) = \ln\left(\sqrt{(x\(-1\))(\(2\)x+\(1\))^{\(2\)}}\right) - \ln(x^{\(2\)}\(-9)
Apply Power Property: Next, we use the property of logarithms that $\ln(a^{n}) = n\cdot\ln(a)$ . Since the square root is the same as raising to the power of $\frac{1}{2}$ , we can apply this property to the first term. $\ln(\sqrt{(x-1)(2x+1)^{2}}) = \frac{1}{2}\cdot\ln((x-1)(2x+1)^{2})$
Combine Terms: Now, we apply the same property to the squared term inside the logarithm. $\newline$ $(\frac{1}{2})\ln((x-1)(2x+1)^{2}) = (\frac{1}{2})(\ln(x-1) + 2\ln(2x+1))$
Write Expanded Form: We can now combine the steps to write the expanded form of the original expression. \ln\left(\frac{\sqrt{(x\(-1\))(\(2\)x+\(1\))^{\(2\)}}}{x^{\(2\)}\(-9\)}\right) = \frac{\(1\)}{\(2\)}\cdot\left(\ln(x\(-1) + $2$ \cdot\ln( $2$ x+ $1$ )\right) - \ln(x^{ $2$ } $-9$ )
Simplify Expression: Finally, we simplify the expression by distributing the $\frac{1}{2}$ and multiplying the $2$ inside the logarithm. $\left(\frac{1}{2}\right)\cdot\left(\ln(x-1) + 2\cdot\ln(2x+1)\right) = \left(\frac{1}{2}\right)\cdot\ln(x-1) + \ln(2x+1) - \ln(x^{2}-9)$

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