Which of the following are equivalent to \( 2 \cdot \log \left(u^{-1 / 3}\right) \) ? Assume \( u>0 \). \( \square-\frac{2}{3} \log (u) \) \( \square \log \left(\frac{1}{\sqrt[3]{u^{2}}}\right) \) \( \square-\frac{1}{3} \log \left(u^{2}\right) \) \( \square-\log \left(u^{2 / 3}\right) \) \( \log \left(\frac{1}{u^{2 / 3}}\right) \)

To explore the expression \( 2 \cdot \log \left(u^{-1 / 3}\right) \), we can first apply the logarithmic property that states \( \log(a^b) = b \cdot \log(a) \). Using this, we rewrite the expression: \[ 2 \cdot \log \left(u^{-1/3}\right) = 2 \cdot \left(-\frac{1}{3} \log(u)\right) = -\frac{2}{3} \log(u). \] Now, let’s check the other options. 1. \( \log \left(\frac{1}{\sqrt[3]{u^{2}}}\right) \) simplifies to: \[ \log \left(u^{-2/3}\right) = -\frac{2}{3} \log(u), \] which is equivalent. 2. \( -\frac{1}{3} \log(u^{2}) = -\frac{1}{3} \cdot 2 \log(u) = -\frac{2}{3} \log(u), \) hence, this one is equivalent too. 3. \( -\log \left(u^{2/3}\right) = -\frac{2}{3} \log(u), \) thus, it is also equivalent. 4. For \( \log \left(\frac{1}{u^{2/3}}\right) \): \[ \log \left(u^{-2/3}\right) = -\frac{2}{3} \log(u), \] confirming it is equivalent. So, all provided options are indeed equivalent to \( 2 \cdot \log \left(u^{-1 / 3}\right) \)!