Self-studying here. What are the steps to express $\ln \frac{8 \cdot 4^{1/3}}{\sqrt 2}$ as $\frac{19}{6}\ln 2$? I know the usual rules for manipulating logs and have tried several times to get the answer I want, but haven't quite gotten there. The actual question is how to express the first figure in terms of $\ln 2$, and the book I have provided the solution. I just haven't been able to determine the steps working forward or backward. Your help appreciated.
[Math] Steps to express $\ln \frac{8 \cdot 4^{1/3}}{\sqrt 2}$ as $\frac{19}{6}\ln 2$
algebra-precalculus
Best Answer
Using the laws of exponents:
$$\frac{8 \cdot 4^{1/3}}{\sqrt 2} = \frac{2^3 \cdot (2^2)^{1/3}}{2^{1/2}} = 2^3 \cdot 2^{2/3} \cdot 2^{-1/2}$$
$$= 2^{3 + 2/3 - 1/2}$$
which simplifies to:
$$= 2^{19/6}$$
Now, using the laws of logarithms, we have: $$\ln (2^{19/6}) = \frac{19}{6} \ln 2$$ And we are done.