Why is log(x) ≤ x - 1 for All x?

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SUMMARY

The inequality log(x) ≤ x - 1 holds true for all x in the domain of real numbers. At x = 1, both log(1) and 1 - 1 equal zero, confirming the equality at this point. Analyzing the derivatives of the functions log(x) and x - 1 reveals that the derivative of log(x) is 1/x, which is always less than or equal to 1 for x > 0, while the derivative of x - 1 is constant at 1. This indicates that log(x) grows slower than x - 1 for all x > 0.

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  • Understanding of logarithmic functions
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esrak23
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Why was logx less than or equal to x-1 for all x?
Thanks in advance for your help!
 
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Compare the value of log(x) and x-1 at x=1.
Then compare the derivatives of these two fubctions and reach an interesting conclusion.
 
oh yeah...that makes sense. thank you!
 

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