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# Solve log(base 10) cube root of 10. Without a calculator?

thanks! :)

### 7 Answers

- CoffinLv 41 decade agoFavorite Answer
This is all a matter of knowing properties of exponents and logarithms.

» ⁿ√x = x¹ / ⁿ

log (³√10) = log (10¹ / ³)

» log (xⁿ) = n log x

log (10¹ / ³) = (1/3) log 10

» log 10 = 1

(1/3)(1) = (1/3)

- OrionLv 41 decade ago
By properties of exponents the cube root of 10 is equivalent to 10^(1/3), Then our statement reads:

log (10^(1/3))

Here I mean log (#) is implicitly log base 10. The issue is not in solving this, but evaluating. There are no unknowns and hence you cannot solve anything. By properties of logarithms the power 1/3 can come out as a coefficient of the logarithm.

Hence, log (10^(1/3)) = (1/3)*log (10). And again by properties of logarithms, log (10) = 1 by the identity property of logarithms.

Hence (1/3)*log (10) = (1/3)*1 = 1/3.

So log (10^(1/3)) = 1/3.

- Anonymous5 years ago
Essentially, this is just a simple logarithm question. Treat the equation as a 'power' equation, and notice that you have a 10 to the power. We can therefore use log to the base 10: x log(6) = log(10) Resulting in: x = log 10 / log 6 The numbers you have chosen are not easy to manipulate without a calculator. It is however possible to use log to the base 6, which ends up with a messier answer.

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