Question

expanding logarithmic expressions
expand each expression.
$ln (2x)^4$
$circ$ $4 ln 2 + 4 ln x$
$circ$ $4 ln 2 + ln x$
$\times$ $8 ln x$
retry

Explanation:

Step1: Apply Power Rule of Logarithms

The power rule of logarithms states that $\ln(a^b) = b\ln(a)$. For the expression $\ln((2x)^4)$, we can apply this rule first. So, $\ln((2x)^4) = 4\ln(2x)$.

Step2: Apply Product Rule of Logarithms

The product rule of logarithms states that $\ln(ab) = \ln(a) + \ln(b)$. Now, we apply this rule to $\ln(2x)$. So, $\ln(2x) = \ln(2) + \ln(x)$.

Step3: Substitute and Simplify

Substitute $\ln(2x) = \ln(2) + \ln(x)$ into $4\ln(2x)$. We get $4(\ln(2) + \ln(x)) = 4\ln(2) + 4\ln(x)$.

Answer:

$4\ln 2 + 4\ln x$ (corresponding to the first option: 4 ln 2 + 4 ln x)