Explained: "For what number n does the equation lognx=nx..."

Milton Shepard

Answered question

2022-04-12

For what number

n

does the equation

\log_{n} x =^{n} x

have exactly one solution?

Gonarsu2dw8

Beginner2022-04-13Added 19 answers

x \to \log n (x)

and

x \to n^{x}

are each other's inverses, so if we're looking for an

n

where they just touch each other, we're looking for an

n

where one of them has the line

x = y

as a tangent
Set

α = \ln n

; then one of the graphs is

f (x) = e^{α x}

. The tangent condition is that for some

x_{0}

we have

f (x_{0}) = x_{0} and f^{'} (x_{0}) = 1

or, concretely

e^{α x_{0}} = x_{0} and α e^{α x_{0}} = 1

Combining these, we find

α x_{0} = 1

, so

x_{0} = \frac{1}{α}

, which we can plug back into the second equation to get

α e^{1} = 1

which implies

α = e^{- 1}

and thus

n = e α = e^{e^{- 1}}

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