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I'm sure this is trivial. I want to reform the basis of a logarithm, so that I transform a factor

10.0/log(10.0)

into a factor

1.0/log(x)

so I want to solve x in

1.0/log(x) = 10.0/log(10.0)

I reciprocate and exponentiate

x = exp(log(10.0)/10.0)
  = exp(log(10.0)) - exp(10.0)  // ?
  = 10.0 - exp(10.0)

But this is wrong; it yields -22016.465 instead of the empirical result 1.258926.

Ok, the answer seems to be $\sqrt[10]{10}$. But why?

Emit Taste
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1 Answers1

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As mentioned in the comments, you made a mistake when you converted $e^{\frac{\ln 10}{10}}$ into $e^{\ln 10}-e^{10}$. This is incorrect; exponents don't work that way like logarithms do.

Instead, what you should do is notice that

$$\frac{\ln 10}{10}=\frac{1}{10} \ln 10=\ln({10^\frac{1}{10}})=\ln \sqrt[10]{10}$$

and then you have

$$e^{\frac{\ln 10}{10}}=e^{(\ln \sqrt[10]{10})}=\sqrt[10]{10}$$

Kman3
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