Problem
ALG-B2-M05-P015 Logarithmic tangent
#15
★★★★☆ Level 4 of 5
Prove for \(x>0\): \[\ln x\le x-1.\]
Hint. Use the tangent to \(\ln x\) at \(1\).
The function \(\ln x\) is concave, so its graph lies below every tangent. At \(1\), the tangent has equation \(y=x-1\). Hence \(\ln x\le x-1\).
This basic tangent is often used for products.