Question

base e
expanding expressions
expand: $ln\frac{2b^{4}}{a^{3}}$
$ln 2 - 3ln a+4ln b$
$ln 2 + 3ln a-4ln b$
$ln 2 - 3ln a-4ln b$
ccsd pre - calculus cp ic 25 - 26

Explanation:

Step1: Apply log - quotient rule

The quotient rule for logarithms is $\ln\frac{M}{N}=\ln M-\ln N$. So, $\ln\frac{2b^{4}}{a^{3}}=\ln(2b^{4})-\ln(a^{3})$.

Step2: Apply log - product rule

The product rule for logarithms is $\ln(MN)=\ln M+\ln N$. Then $\ln(2b^{4})=\ln2+\ln(b^{4})$.

Step3: Apply log - power rule

The power rule for logarithms is $\ln(M^{n}) = n\ln M$. So, $\ln(b^{4}) = 4\ln b$ and $\ln(a^{3})=3\ln a$.
Combining these results: $\ln\frac{2b^{4}}{a^{3}}=\ln2 + 4\ln b-3\ln a=\ln2-3\ln a + 4\ln b$.

Answer:

$\ln2-3\ln a + 4\ln b$