# Steps to express $\ln \frac{8 \cdot 4^{1/3}}{\sqrt 2}$ as $\frac{19}{6}\ln 2$

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.

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.

• This is proof that I must practice basics after learning them and continuing on in mathematics. To this end, I am now using Anki to make sure I don't forget these elementary facts, like logarithmic and exponential manipulation. If you don't use it, you lose it unfortunately. Getting more sleep could help too. – Joe May 4 '15 at 17:38