Question

nding logarithmic expressions
expand the expression.
$ln (2x)^4$
$4ln 2 + 4ln x$ $8ln x$ $4ln 2 + ln x$

Explanation:

Step1: Apply the 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:
$\ln((2x)^4) = 4\ln(2x)$

Step2: Apply the product rule of logarithms

The product rule of logarithms states that $\ln(ab) = \ln(a) + \ln(b)$. Now we apply this rule to $\ln(2x)$:
$4\ln(2x) = 4(\ln(2) + \ln(x))$

Step3: Distribute the 4

Distribute the 4 across the sum:
$4(\ln(2) + \ln(x)) = 4\ln(2) + 4\ln(x)$

Answer:

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