TAOCP 1.2.2 Exercise 30

Section 1.2.2: Numbers, Powers, and Logarithms

Exercise 30. [12] Simplify the expression $(\ln x)^{\ln x / \ln \ln x}$, assuming that $x > 1$ and $x \ne e$.

Verified: no
Solve time: 19s

Solution

By equation (15),

$$ \ln x = \log_e x. $$

Hence

$$ (\ln x)^{\ln x / \ln \ln x} = \left(e^{\ln \ln x}\right)^{\ln x / \ln \ln x}. $$

Using the law of exponents (5),

$$ \left(e^{\ln \ln x}\right)^{\ln x / \ln \ln x} = e^{(\ln \ln x)(\ln x / \ln \ln x)} = e^{\ln x}. $$

By equation (9),

$$ e^{\ln x} = x. $$

Therefore the given expression simplifies to

$$ \boxed{x}. $$

The assumptions $x>1$ and $x\ne e$ ensure that $\ln x>0$ and $\ln\ln x\ne0$, so the exponent is defined. ∎