Logarithm and Factorial Relation

Calculus Level 2

If x > 1 x>1 and is a whole number, what sign should be inside the box? log ⁡ x ( x ! ) ? x \large \log_{x}(x!)\space \small \boxed{?} \space \large x

> > = = Varies depending on the value of x x < <

Kaizen Cyrus by Kaizen Cyrus , 18, Philippines

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1 solution

Parth Sankhe
Feb 4, 2019

Since 1 × 2 × 3 × . . . × x < x × x × x . . . × x 1×2×3×...×x < x×x×x...×x , i.e. x ! < x x x!<x^x ,

log ⁡ x x ! < log ⁡ x x x \log_x x!<\log_x x^x

Thus, log ⁡ x x ! < x \log_x x!<x

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