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Say I have a number with a fractional exponent, $10^{\frac{1}{3}}$. Could this number be considered a logarithm, even though it is not written as $10^{0.\overline{3}}$?

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$10^{\frac13}=\sqrt[3]{10}$ and $\log_ab = c \iff a^c = b$ –  axblount Sep 14 '12 at 15:58
    
@axblount, I know that. –  Dan the Man Sep 14 '12 at 15:59
    
Yes: $\ln e^{10^{1/3}}=10^{1/3}$. –  Raskolnikov Sep 14 '12 at 16:04
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Why does it matter how the exponent is written? $\tfrac{1}{3}$ and $0.\overline{3}$ are two notations representing exactly the same number: the unique real number which yields 1 when tripled. However, $10^{1/3}$ is not really "a logarithm", except in the sense that Raskolnikov indicates (that it is the logarithm of some other number, such as $\mathrm{e}^{10^{1/3}}$). –  Niel de Beaudrap Sep 14 '12 at 16:06
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@DantheMan, Sorry for the glib comment. I see what you're saying. $\log$ asks "what power should I raise this number to". $\sqrt{}$ asks "what number should I raise to this power". They are similar in a sense. –  axblount Sep 14 '12 at 16:26

2 Answers 2

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The question "What is $\log_bx$?" is equivalent to "What power do I need to raise $b$ to in order to get $x$?"

The question "What is $\sqrt[n]{x}$ ($=x^{\frac1n}$)?" is equivalent to "What number should I raise to the power $n$ in order to get $x$?"

They are both questions about an equation of the form $a^b=c$. That being said, I wouldn't call $10^\frac13$ a logarithm.

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$10^{\frac{1}{3}}$ and indeed $x^{\frac{1}{n}}$ for integer $n\gt 1$ can be called a root, as they are $\sqrt[3]{10}$ or $\sqrt[n]{x}$.

Looking at $10^{\frac{1}{3}}$, you can say that $\frac{1}{3}$ is its (base $10 \,$ or common) logarithm.

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