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

Update:

thanks! :)

7 Answers

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  • Coffin
    Lv 4
    1 decade ago
    Favorite 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)

  • Orion
    Lv 4
    1 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.

  • Anonymous
    5 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.

  • 4 years ago

    Cube Root Of 10

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  • frank
    Lv 7
    1 decade ago

    log(base 10) (10)^1/3

    = 1/3 log(base 10) 10

    = 1/3 (1)

    = 1/3 (answer)

  • 1 decade ago

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

  • Fred
    Lv 7
    1 decade ago

    log10(10^ 1/3) = 1/3 log10(10) = 1/3

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