Logarithmic functions are central in calculus because they describe growth that slows down as the input increases. The natural logarithm, written ln x, has a simple and powerful derivative: d/dx ln x = 1/x. This rule appears in physics, chemistry, economics, biology, and any setting where ratios, scales, or exponential relationships are studied.
Understanding it also helps connect logarithms and exponentials as inverse functions.
Key Facts
- Natural log rule: d/dx ln x = 1/x, for x > 0.
- General log rule: d/dx log_a x = 1/(x ln a), where a > 0 and a != 1.
- Chain rule for natural logs: d/dx ln u = u'/u, where u is a differentiable function of x.
- Power inside a log: d/dx ln(x^n) = n/x for x > 0, which matches ln(x^n) = n ln x.
- Log of a product: ln(ab) = ln a + ln b, so logarithmic differentiation can turn products into sums before differentiating.
- The slope of y = ln x is positive but decreases as x increases because 1/x gets smaller.
Vocabulary
- Natural logarithm
- The logarithm with base e, written ln x, where e is approximately 2.718.
- Derivative
- A derivative gives the instantaneous rate of change or slope of a function at a point.
- Chain rule
- The chain rule is a differentiation rule used when one function is inside another function.
- Logarithmic differentiation
- Logarithmic differentiation is a method that takes the logarithm of both sides to make complicated products, quotients, or powers easier to differentiate.
- Domain
- The domain is the set of input values for which a function is defined, and ln x requires x > 0.
Common Mistakes to Avoid
- Writing d/dx ln x = x is wrong because the derivative of ln x is the reciprocal 1/x, not the original input.
- Forgetting the chain rule in ln(3x^2 + 1) is wrong because the inside function must also be differentiated, giving 6x/(3x^2 + 1).
- Using d/dx log_a x = 1/x for every base is wrong because only the natural logarithm has derivative 1/x; other bases require the factor 1/ln a.
- Ignoring the domain of logarithms is wrong because ln x and log_a x are defined only for positive inputs in real-valued calculus.
Practice Questions
- 1 Find d/dx ln(5x) and simplify your answer.
- 2 Find the derivative of f(x) = ln(2x^2 + 3x) at x = 2.
- 3 Explain why the tangent slope of y = ln x is steeper at x = 1 than at x = 5.