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Whether measuring earthquake intensities on the Richter scale or sound levels in decibels, nature often moves on logarithmic scales. But how do we measure the instantaneous rate of change of a scale that squishes giant numbers into manageable values?
It turns out the rate of change of the simplest logarithm produces one of the cleanest results in all of calculus.

The Natural Logarithm Rule

The natural logarithm, written as ln⁡(x)\ln(x), is the logarithm with base ee, where Euler's number e≈2.71828e \approx 2.71828. For all x>0x > 0, its derivative is simply the reciprocal function: ddx[ln⁡(x)]=1x\frac{d}{dx}[\ln(x)] = \frac{1}{x}
In 1614, Scottish mathematician John Napier invented logarithms to turn painful multiplication into easy addition. We can prove this derivative directly using implicit differentiation: if y=ln⁡(x)y = \ln(x), rewriting in exponential form gives ey=xe^y = x.
Differentiating both sides with respect to xx yields ey⋅dydx=1e^y \cdot \frac{dy}{dx} = 1, which means dydx=1ey=1x\frac{dy}{dx} = \frac{1}{e^y} = \frac{1}{x}.
What happens if we scale the variable inside by a constant factor, like ln⁡(5x)\ln(5x)? Does the slope get 5 times steeper?

The Mystery of ln(kx)

Let's test f(x)=ln⁡(kx)f(x) = \ln(kx), where kk is any positive constant. Applying the chain rule, we differentiate the outer logarithm to get 1kx\frac{1}{kx} and multiply by the derivative of the inside, which is kk.
f′(x)=1kx⋅k=1xf'(x) = \frac{1}{kx} \cdot k = \frac{1}{x} The constant kk cancels out completely, leaving the exact same derivative as ln⁡(x)\ln(x)!
A common mistake is claiming that ddx[ln⁡(7x)]=7x\frac{d}{dx}[\ln(7x)] = \frac{7}{x} by accidentally leaving the inner derivative in the numerator without simplifying. Always remember: the constant inside the argument cancels against the denominator, leaving strictly 1x\frac{1}{x}.