AP Calculus AB and BC

Derivative of log x: Answer, Base Change, Mistakes

The derivative of log base 10 of x with respect to x is 1 over x times ln 10. Change of base rewrites the logarithm as ln x divided by ln 10, and since ln 10 is a constant the derivative is one over x scaled by that constant. The natural logarithm is the only base with no extra factor.

ddx[log⁡10x]=1xln⁡10\frac{d}{dx}\left[\log_{10} x\right] = \frac{1}{x\ln 10}

How to differentiate log base 10 of x

Change of base turns any logarithm into a constant multiple of a natural logarithm.

log⁡10x=ln⁡xln⁡10\log_{10} x = \frac{\ln x}{\ln 10}

ln⁡10\ln 10 is just a number, so the constant multiple rule pulls it out and the derivative of ln⁡x\ln x finishes the job.

ddxlog⁡ax=1xln⁡a\frac{d}{dx}\log_{a} x = \frac{1}{x\ln a}

Why base e is the natural choice

Setting a=ea = e makes ln⁡a=1\ln a = 1, so the extra factor disappears and the derivative is simply 1x\frac{1}{x}. Every other base carries a constant along for the ride.

That is what "natural" means here: it is the one base whose logarithm differentiates cleanly, which is why calculus uses it almost exclusively.

Where the derivative of log x shows up on the AP exam

Topic 2.8 on both exams. Notation matters: on the AP exam log⁡x\log x conventionally means base 1010 and ln⁡x\ln x means base ee, so read the notation before differentiating.

Common mistakes with the derivative of log x

  • Answering 1x\frac{1}{x}, which is the derivative of ln⁡x\ln x, not of log⁡10x\log_{10}x.
  • Putting the constant in the wrong place and writing ln⁡10x\frac{\ln 10}{x}. The ln⁡10\ln 10 divides, so it belongs in the denominator.
  • Writing 1xlog⁡10\frac{1}{x\log 10}. Change of base gives a NATURAL logarithm in the denominator, and log⁡1010\log_{10}10 would just be 11.

Check yourself, not just the answer

Type derivatives and get graded on mathematical equivalence, with rule-level hints when you miss, in the Derivative Practice Checker.

Frequently asked questions

What is the derivative of log x?

With log⁡\log meaning base 1010, it is 1xln⁡10\frac{1}{x\ln 10}.

What is the derivative of log base a of x?

It is 1xln⁡a\frac{1}{x\ln a}, from change of base.

Why is the derivative of ln x simpler?

Because ln⁡e=1\ln e = 1, so the constant factor becomes 11 and vanishes. That is what makes base ee natural.

Does log mean base 10 or base e on the AP exam?

log⁡\log without a stated base means base 1010; the natural logarithm is always written ln⁡\ln.