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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.)

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  • $\begingroup$ Can you write $10/\root3\of {10}$ as $10^x$ for some $x$? If so, what would $\log 10^x$ be? $\endgroup$ – David Mitra Jul 29 '14 at 10:39
  • $\begingroup$ @DavidMitra I got as far as $10(10^{-\frac13})$ but don't know where to go from there $\endgroup$ – imulsion Jul 29 '14 at 10:40
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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?

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  • $\begingroup$ Thanks for your answer, but I don't understand why $\frac{10}{10^{1/3}} = 10^{1-1/3}$. Probably me being a moron, but I don't understand how you got there. $\endgroup$ – imulsion Jul 29 '14 at 10:46
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    $\begingroup$ @imulsion The following rule is very good to know: $\frac{x^a}{x^b} = x^{a-b}$. Let $x=10$, $a=1$ and $b=1/3$. (Remember that $10 = 10^1$.) $\endgroup$ – naslundx Jul 29 '14 at 10:48
  • $\begingroup$ I can't believe that I forgot that :facepalm:. Thanks very much for your help though, I will accept your answer when I can $\endgroup$ – imulsion Jul 29 '14 at 10:49
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    $\begingroup$ @imulsion No need to feel stupid, we are all here to learn. :) $\endgroup$ – naslundx Jul 29 '14 at 10:50
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Remember that the logarithm of a quotient is the difference of logarithms: $$log\left(\frac{10}{\sqrt[3]{10}}\right)=log(10)-log(\sqrt[3]{10})=1-log(10^{1/3})=1-\frac {1}{3}\cdot log(10)=1-\frac {1}{3}=\frac {2}{3}$$

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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 $\log_{10}\frac{10}{\sqrt[3]{10}}=\frac{2}{3}$

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