Questions: Use logarithmic identities to expand each expression, rewriting it as a sum, difference, or multiple of logarithms. [ ln left(frace5 x^3right)= ]

Solution Steps

To expand the given logarithmic expression using logarithmic identities, we can apply the following rules: the logarithm of a quotient is the difference of the logarithms, and the logarithm of a power is the exponent times the logarithm of the base. Specifically, we will use these identities to separate the terms inside the logarithm.

We start with the expression \( \ln \left( \frac{e}{5 x^{3}} \right) \). Using the logarithmic identity for the quotient, we can rewrite this as: \[ \ln(e) - \ln(5 x^{3}) \]

Next, we apply the logarithmic identity for the product inside the logarithm: \[ \ln(5 x^{3}) = \ln(5) + \ln(x^{3}) \] Using the power rule for logarithms, we can express \( \ln(x^{3}) \) as \( 3 \ln(x) \). Thus, we have: \[ \ln(5 x^{3}) = \ln(5) + 3 \ln(x) \]

Substituting back into our expression, we get: \[ \ln(e) - \left( \ln(5) + 3 \ln(x) \right) \] Since \( \ln(e) = 1 \), we can simplify this to: \[ 1 - \ln(5) - 3 \ln(x) \]

Final Answer