Derivative of log(x)
Base-10 logs differentiate almost like natural logs — with a constant factor of ln 10 that trips up anyone who assumed all logarithms behave the same.
Step-by-step solution
Change of base: log(x) = ln(x) / ln(10). The denominator ln 10 ≈ 2.3026 is just a constant.
d/dx[ln(x)] = 1/x, and the constant 1/ln(10) rides along.
f′(x) = 1/(x ln 10) ≈ 0.4343/x.
Why it works
The factor 1/ln(10) is the price of using a base other than e. The natural log is "natural" precisely because it's the one base whose derivative has no correction constant — every other base b gives 1/(x ln b). Using base e makes the natural-log derivative especially simple.
One caution about notation: in pure math and in many programming languages, log with no subscript means the natural log, while in engineering, chemistry (pH), and most school textbooks it means base 10. DerivCalc uses the base-10 convention — log is base 10, ln is base e — but always check which convention your course uses before trusting any formula sheet.
Base conventions and domain
On DerivCalc, log(x) means log₁₀(x), while ln(x) means the natural logarithm. Both real-valued functions require x > 0. Their derivative formulas apply on this same positive domain.
For a general logarithm base b, require b > 0 and b ≠ 1. Change of base gives logb(x) = ln(x)/ln(b), so its derivative is 1/(x ln(b)). If 0 < b < 1, ln(b) is negative and the logarithm is decreasing.
For a composite base-10 logarithm, d/dx[log(g(x))] = g′(x)/(g(x) ln(10)) where g is differentiable and g(x) > 0. For log(3x + 2), the result is 3/((3x + 2) ln(10)) with x > −2/3. Check the base convention in your course before comparing answers.
Common mistakes
- Answering 1/x and dropping the ln 10 — the most common error, caused by pattern-matching on the natural log result.
- Putting ln 10 in the numerator instead of the denominator.
- Assuming your calculator, your textbook, and this site all mean the same thing by "log" — conventions differ, and the derivative differs with them.
Practice problems
Differentiate log(x) · ln(10)
Answer: 1/x — the constants cancel perfectly.
Differentiate 2 log(x)
Answer: 2/(x ln 10).
Which is larger at x = 5: the slope of ln(x) or of log(x)?
Answer: ln(x): 1/5 = 0.2 versus 1/(5 ln 10) ≈ 0.087.