ln((1)/(x))+ln(2x^(3))=ln 3

Combine logarithms: We are given the equation ln((1)/(x)) + ln(2x^(3)) = ln 3. We will use the properties of logarithms to combine the left side of the equation into a single logarithm.Simplify combined expression: Using the property of logarithms that ln(a) + ln(b) = ln(a*b), we can combine the two logarithms on the left side of the equation: ln((1)/(x)) + ln(2x^(3)) = ln((1/x) * 2x^(3)).Cancel x terms: Simplify the expression inside the logarithm: ln((1/x) * 2x^(3)) = ln(2x^(3)/x).Equating arguments: Simplify the expression further by canceling x in the numerator and the denominator: ln(2x^(3)/x) = ln(2x^(2)).Isolate x^2: Now we have the equation ln(2x^(2)) = ln 3. Since the natural logarithm function ln(x) is one-to-one, we can equate the arguments of the logarithms: 2x^(2) = 3.Take square root: Divide both sides of the equation by 2 to isolate x^(2): x^(2) = 3/2.Take the square root of both sides to solve for x. Remember that taking the square root gives us both positive and negative solutions: x = ±√(3/2).

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$\ln\left(\frac{1}{x}\right)+\ln(2x^{3})=\ln 3$

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Algebra 1

Multiplication with rational exponents

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\ln\left(\frac{1}{x}\right)+\ln(2x^{3})=\ln 3

Combine logarithms: We are given the equation $\ln\left(\frac{1}{x}\right) + \ln(2x^{3}) = \ln 3$ . We will use the properties of logarithms to combine the left side of the equation into a single logarithm.
Simplify combined expression: Using the property of logarithms that $\ln(a) + \ln(b) = \ln(a*b)$ , we can combine the two logarithms on the left side of the equation: $\newline$ $\ln\left(\frac{1}{x}\right) + \ln(2x^{3}) = \ln\left(\frac{1}{x} * 2x^{3}\right)$ .
Cancel $x$ terms: Simplify the expression inside the logarithm: $\ln\left(\frac{1}{x} \cdot 2x^{3}\right) = \ln\left(\frac{2x^{3}}{x}\right)$ .
Equating arguments: Simplify the expression further by canceling $x$ in the numerator and the denominator: $\ln(\frac{2x^{3}}{x}) = \ln(2x^{2})$ .
Isolate $x^2$ : Now we have the equation $\ln(2x^{2}) = \ln 3$ . Since the natural logarithm function $\ln(x)$ is one-to-one, we can equate the arguments of the logarithms: $\newline$ $2x^{2} = 3$ .
Take square root: Divide both sides of the equation by $2$ to isolate $x^{2}$ : $\newline$ $x^{2} = \frac{3}{2}$ .
Take square root: Divide both sides of the equation by $2$ to isolate $x^{2}$ : $x^{2} = \frac{3}{2}$ .Take the square root of both sides to solve for $x$ . Remember that taking the square root gives us both positive and negative solutions: $x = \pm\sqrt{\frac{3}{2}}$ .