Common logarithm question

Question

I'm studying logarithms and am doing an exercise where you're supposed to evaluate the solutions of common logarithms without using a calculator. I'm very stuck on this one particular question. I know the answer because I used my calculator, but I'd like to know how to solve it without one. The question is $$\log\left(\frac{10}{\sqrt[\large3]{10}}\right)$$

How do I solve this without a calculator? (Please provide a step-by-step solution, this has really confused me.)

Answer 1

Hint

$$\frac{10}{\sqrt[3]{10}} = \frac{10}{10^{1/3}} = 10^{1-1/3} = 10^{2/3}$$

Now, what would the logarithm (assuming base 10) of that final expression be?

Answer 2

Remember that the logarithm of a quotient is the difference of logarithms: $$log\left(\frac{10}{\sqrt[3]{10}}\right)=log(10)-log(\sqrt[3]{10})=1-log(10^{1/3})=1-\frac {1}{3}\cdot log(10)=1-\frac {1}{3}=\frac {2}{3}$$

Answer 3

To get a good understanding of logarithms it is good to realize that the following two questions are equivalent:

1) What is the logarithm of $a$ on base of $g$? I.e. $\log$$_{g}\left(a\right)=?$.

2) To what power must $g$ be raised to get $a$ as outcome? I.e. $g^{?}=a$.

Here $a>0$, $g>0$ and $g\neq1$.

So $10^{\frac{2}{3}}=\frac{10}{\sqrt[3]{10}}$ is the same information as $\log_{10}\frac{10}{\sqrt[3]{10}}=\frac{2}{3}$