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According to Google calculator, $2\ln(x) = \frac{\ln(x)}{5}$ for many values of $x$. As I remember my logarithm rules, I don't understand why this should be. Can anyone explain?

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Can you add a link of what google calculator displayed? – user17762 Feb 7 '13 at 3:05
This equation is equivalent to $10\ln x = \ln x$, and it is equivalent to $x^{10}=x$, $x>0$. It has only one solution $x=1$, if $x$ is real. But you admit complex solutions, It has infinitely many solutions, namely $x=2n\pi i$, $n\in\mathbb{Z}$. – Hanul Jeon Feb 7 '13 at 3:08
This just implies $\sqrt(x)=x^{\frac{1}{5}}$. This has only two real solutions $1$ and $0$. I think there must be a bunch of complex solutions too. Maybe even infinite. – AvZ Jan 30 '15 at 16:11

Are you sure that is not .2? This could be the source of the misunderstanding.

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I was flipping back and forth and I didn't notice that the decimal point had moved! Sorry, can't figure out how to delete my question. – user61316 Feb 7 '13 at 3:10
Don't delete it. Better, add and Edit to your question adding this little thing. – DonAntonio Feb 7 '13 at 3:16

$$ 2a = 10\left(\frac a5\right). $$

When you multiply by $10$, you move the decimal point over one place to the right.

For example, suppose $a = 53.14$. Then $$ 2a = 106.28 \text{ and } \frac a5 = 10.628. $$ The difference between $106.28$ and $10.628$ is the location of the decimal point.

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