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 $10^{0.\overline{3}}$?
Are fractional exponents considered logarithms?
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0$10^{\frac13}=\sqrt[3]{10}$ and $\log_ab = c \iff a^c = b$ – 2012-09-14
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0@axblount, I know that. – 2012-09-14
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0Yes: $\ln e^{10^{1/3}}=10^{1/3}$. – 2012-09-14
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1Why 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 $\mathrm{e}^{10^{1/3}}$). – 2012-09-14
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0To call a constant "a logarithm" seems like a pointless, counterproductive thing to do. Most of the real line can be written as the logarithm of whatever (positive) base you like! Now if you want to shorten "logarithmic function" to "a logorithm", that seems like a fine thing to do, but those are functions, not constants. – 2012-09-14
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1@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. – 2012-09-14
2 Answers
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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.