Simplification of Terms $ 2^{(1/\ln2)}$

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The final answer is $\mathrm e$. Can anyone show steps? $ 2^{(1/\ln2)}$

I tried log laws without any result. I'm pretty sure I'm stuck somewhere really easy.

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4 Answers

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Try this: $1/ln2 = lne/ln2$ which using the base conversion formula is $log_2e$ thus you get $2^{log_2e}$ which is exactly the same as simply $e$.

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Here's another way to deal with this, using the "change-of-base" formula:

From

$$ \log_b \ x \ = \ \frac{\log_a \ x}{\log_a \ b} \ \ , $$

we have

$$ \ln \ 2 \ = \ \log_e \ 2 \ = \ \frac{\log_2 \ 2}{\log_2 \ e} \ = \ \frac{1}{\log_2 \ e} \ \ . $$

Thus,

$$ 2^{1 / \ln 2} \ = \ 2^{\log_2 \ e \ } \ = \ e \ \ . $$

[We can see that there is a general relation $ \log_b \ a \ = \ \frac{1}{\log_a \ b} \ \ . $ ]

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$$2^{\frac{1}{\ln2}}=x$$ $$\ln\left(2^{\frac{1}{\ln2}}\right)=\ln x$$ $$\frac{1}{\ln2}\cdot\ln2=\ln x$$ $$1=\ln x$$ $$x=e^1=e$$

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Hint: The general definition of $a^b$ is via the exponential function: $a^b:=\exp(b\ln a)$.

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