To determine which of the following expressions is equivalent to [tex]\(4 \ln (3 x)\)[/tex] for [tex]\(x > 0\)[/tex], let's break down the given expression using logarithmic properties step-by-step.
1. Given Expression:
[tex]\(4 \ln (3 x)\)[/tex]
2. Using the Power Rule of Logarithms:
The power rule [tex]\(a \ln b = \ln (b^a)\)[/tex] can be applied to simplify the expression:
[tex]\[
4 \ln (3 x) = \ln \left((3 x)^4\right)
\][/tex]
3. Expand the Argument:
Next, expand [tex]\((3 x)^4\)[/tex]:
[tex]\[
(3 x)^4 = 3^4 \cdot x^4 = 81 \cdot x^4
\][/tex]
4. Simplify the Logarithm:
Substitute back into the logarithm:
[tex]\[
\ln \left((3 x)^4\right) = \ln (81 x^4)
\][/tex]
Now we have:
[tex]\[
4 \ln (3 x) = \ln (81 x^4)
\][/tex]
We see that the expression [tex]\(\ln (81 x^4)\)[/tex] exactly matches one of the provided options. Therefore, the correct answer is:
[tex]\[
\boxed{\ln \left(81 x^4\right)}
\][/tex]
5. Verification against Options:
- [tex]\(\ln 81 + \ln x\)[/tex]
- [tex]\(\ln (12 x)\)[/tex]
- [tex]\(\ln 12 + \ln x\)[/tex]
- [tex]\(\ln \left(81 x^4\right)\)[/tex]
The expression equivalent to [tex]\(4 \ln (3 x)\)[/tex] matches [tex]\(\ln \left(81 x^4\right)\)[/tex].
Thus, the correct answer is:
[tex]\[
4 \ln (3 x) \equiv \ln \left(81 x^4\right)
\][/tex]
Choose the correct answer below:
[tex]\(
\ln \left(81 x^4\right)
\)[/tex]
