ln(lnx)=0?

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embizze | High School Teacher | (Level 1) Educator Emeritus

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Solve ln(ln(x))=0:

Use the fact that ln(x) and e^x are inverse functions: in particular, e^(ln(u))=u.

ln(ln(x))=0  Raise both sides with base e:

e^(ln(ln(x)))=e^0  a^0=1 for all a not zero, so:

ln(x)=1  Raise both sides with base e:

e^(ln(x))=e^1

x=e

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The solution is e where e is approximately 2.7182818284590; e is the base of the natural logarithm.

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Check: ln(ln(e))=ln(1)=0.

The graph of ln(ln(x)):

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Note that the zero is at approximately 2.7 as expected.

Sources:

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