Problem 9 (Logarithms)

Find the positive real solutions of the equation

log_a x = x

where

a = x ^log4 x

Solution
Using the exponential form of the logarithm, we can write

a ^x = x

or

(x ^log4 x ) ^x = x

from which we can write

x ^{x log}4 x = x

Setting the exponents of in this equation equal, we have

x log4 x = 1

or

log4 x ^x = 1

and from the definition of the logarithm, we have x ^x = 4, or x = 2.

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Last modified on Tuesday, January 12, 1999