# Common logarithm 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((10)/(root[3](10))) How do I solve this without a calculator? (Please provide a step-by-step solution, this has really confused me.)

hotonglamoz 2022-09-24 Answered
Common logarithm 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
$\mathrm{log}\left(\frac{10}{\sqrt[3]{10}}\right)$
How do I solve this without a calculator? (Please provide a step-by-step solution, this has really confused me.)
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## Answers (2)

Madden Huber
Answered 2022-09-25 Author has 12 answers
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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Zack Chase
Answered 2022-09-26 Author has 3 answers
Remember that the logarithm of a quotient is the difference of logarithms:
$log\left(\frac{10}{\sqrt[3]{10}}\right)=log\left(10\right)-log\left(\sqrt[3]{10}\right)=1-log\left({10}^{1/3}\right)=1-\frac{1}{3}\cdot log\left(10\right)=1-\frac{1}{3}=\frac{2}{3}$
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