ln((2*b^(-2))/(c^(3)))

Break down using properties: Apply the logarithm properties to break down the expression. The natural logarithm of a quotient is equal to the difference of the natural logarithms of the numerator and the denominator. ln((2*b^(-2))/(c^(3))) = ln(2*b^(-2)) - ln(c^(3))Separate constants and variables: Apply the logarithm properties to separate the constants and variables in the numerator. The natural logarithm of a product is equal to the sum of the natural logarithms of the factors. ln(2*b^(-2)) = ln(2) + ln(b^(-2))Apply power rule: Apply the power rule for logarithms to the terms with exponents. The natural logarithm of a number raised to a power is equal to the power times the natural logarithm of the number. ln(b^(-2)) = -2*ln(b) ln(c^(3)) = 3*ln(c)Substitute and simplify: Substitute the results from Step 3 into the expression from Step 1. ln((2*b^(-2))/(c^(3))) = ln(2) + (-2*ln(b)) - (3*ln(c))Simplify the expression by combining like terms. ln((2*b^(-2))/(c^(3))) = ln(2) - 2*ln(b) - 3*ln(c)

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$\ln\left(\frac{2b^{-2}}{c^{3}}\right)$

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Algebra 1

Multiplication with rational exponents

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\ln\left(\frac{2b^{-2}}{c^{3}}\right)

Break down using properties: Apply the logarithm properties to break down the expression. $\newline$ The natural logarithm of a quotient is equal to the difference of the natural logarithms of the numerator and the denominator. $\newline$ $\ln\left(\frac{2b^{-2}}{c^{3}}\right) = \ln(2b^{-2}) - \ln(c^{3})$
Separate constants and variables: Apply the logarithm properties to separate the constants and variables in the numerator. $\newline$ The natural logarithm of a product is equal to the sum of the natural logarithms of the factors. $\newline$ $\ln(2\cdot b^{-2}) = \ln(2) + \ln(b^{-2})$
Apply power rule: Apply the power rule for logarithms to the terms with exponents. $\newline$ The natural logarithm of a number raised to a power is equal to the power times the natural logarithm of the number. $\newline$ $\ln(b^{-2}) = -2\cdot\ln(b)$ $\newline$ $\ln(c^{3}) = 3\cdot\ln(c)$
Substitute and simplify: Substitute the results from Step $3$ into the expression from Step $1$ . $\newline$ $\ln\left(\frac{2b^{-2}}{c^{3}}\right) = \ln(2) + (-2\ln(b)) - (3\ln(c))$
Substitute and simplify: Substitute the results from Step $3$ into the expression from Step $1$ . $\newline$ $\ln\left(\frac{2b^{-2}}{c^{3}}\right) = \ln(2) + (-2\ln(b)) - (3\ln(c))$ Simplify the expression by combining like terms. $\newline$ $\ln\left(\frac{2b^{-2}}{c^{3}}\right) = \ln(2) - 2\ln(b) - 3\ln(c)$