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Differentiate log x

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Abualtamas

Lvl 1
11y ago
Updated: 4/28/2022

The derivative of ln x, the natural logarithm, is 1/x.

Otherwise, given the identity logbx = log(x)/log(b), we know that the derivative of logbx = 1/(x*log b).

ProofThe derivative of ln x follows quickly once we know that the derivative of ex is itself.

Let y = ln x (we're interested in knowing dy/dx)

Then ey = x

Differentiate both sides to get ey dy/dx = 1

Substitute ey = x to get x dy/dx = 1, or dy/dx = 1/x.

Differentiation of log (base 10) x

log (base 10) x

= log (base e) x * log (base 10) e

d/dx [ log (base 10) x ]

= d/dx [ log (base e) x * log (base 10) e ]

= [log(base 10) e] / x

= 1 / x ln(10)

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