Expressing $\ln \sqrt[3]{54}$ in terms of $\ln 2$ and/or $\ln 3$

Question

Express $\ln \sqrt[3]{54}$ in terms of $\ln 2$ and/or $\ln 3$

I know that $\sqrt[3]{54}=54^{1/3}$ but otherwise I don't know how to address these types of problems. How do I solve this, and is there a general way of tackling these problems? Like, if I had $\ln.75$ or $\ln \frac{8}{9}$, for example. These types of problems that I have are all different and I don't understand a general way of looking at them.

thanks.

Answer 1

There are three properties to know:

• $\ln(ab) = \ln a + \ln b$

• $\ln \left(\frac a b\right) = \ln a - \ln b$

• $\ln(a^b) = b \ln a$

The rest is just putting things together, and using these multiple times. In particular, you'll probably find it useful to know that $54 = 2 \cdot 3^3$; now expand a few times.

Answer 2

As $\displaystyle \ln a^m=m\ln a\ \ \ \ (1)$ and $\ln a+\ln b=\ln(ab)\ \ \ \ (2)$ when logarithms remain defined

Now, $\displaystyle\sqrt[3]{54}=54^{\frac13}$ use $(1)$

Again, $\displaystyle54=2\cdot3^3$ Now use $(2)$