Daniel J Smith Custom TMUA Set A Paper 1 · Question 3

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Daniel J Smith Custom TMUA Set A Paper 1 · Q3

Paper 1Powers & logarithmsAlgebra

Fix $a>0$ with $a\neq1$

For $x>0$ consider the equation

\log_a x=x

The number of positive solutions of this equation is

Aexactly one if

0<a<1

, two if

1<a<e^{1/e}

, one if

a=e^{1/e}

, and none if

a>e^{1/e}

Bexactly two if

0<a<1

, one if

a=1

, and none if

a>1

Cexactly one for all

a>0

a\neq1

Dnone if

0<a<1

, one if

a=e^{1/e}

, and two if

a>e^{1/e}

Enone if

a>1

, and exactly one if

0<a<1

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