MTH6131 Chap.7 Logarithmic and Exponential Derivatives
Logarithmic and Exponential Derivatives
The number e, two ways
The notes recall e as the limit of (1 + 1/h) to the power h as h grows, equivalently the limit of (1 + k) to the power 1/k as k tends to zero, and also as the series 1 + 1 + 1/2! + 1/3! and so on. Both give 2.71828 and onward, but the series gets there much faster.
The same limit is the one that produced continuous compounding in the opening lecture.
Logarithms as tools
For a positive base a not equal to 1, y = log base a of x means a to the power y equals x, and the natural logarithm uses base e. The laws turn products into sums, quotients into differences and powers into multiples, and change of base relates any two logarithms.
In this chapter their job is to simplify before differentiating: the log of a square root becomes half a log, and the log of a product becomes a sum of logs, each with a simple derivative.
The four formulas
The derivative of log base a of x is 1/(x ln a), which reduces to 1/x for the natural logarithm. The derivative of a to the power x is a to the power x times ln a, which reduces to e to the power x for base e.
The notes prove the logarithm formula from first principles, rearranging the difference quotient until the definition of e appears, and then get the exponential formula by implicit differentiation.
Logarithmic differentiation
When a function is a product or quotient of several factors, taking the natural log of both sides first turns it into a sum, and differentiating implicitly gives y′/y directly; multiplying by y finishes the job.
When the variable appears in both the base and the exponent, as in x to the power x, the method is not merely convenient but necessary, since neither the power rule nor the exponential rule applies.
What this chapter covers
- 01
The number e as a limit and as a factorial series
- 02
Logarithms and the change of base
- 03
Log laws used before differentiating
- 04
Derivatives of log and exponential functions
- 05
The first-principles proof for the logarithm
- 06
Logarithmic differentiation of products and quotients
- 07
Variable base and variable exponent
- 08
Turning points of log and exponential curves
Worked example · free
Variable base, variable exponent
- 1Take natural logarithms and simplify with the power law.
- 2Differentiate implicitly using the product rule.
- 2Multiply by y and solve y′ = 0.
Key terms
- Natural Logarithm
- The logarithm to base e, whose derivative is one over x and which undoes the exponential function e to the power x.
- Change of Base
- The identity that the logarithm of u to base a equals the logarithm of u to any base k divided by the logarithm of a to base k.
- Exponential Function
- A function of the form a to the power x with a positive constant base; its derivative is proportional to itself.
- Logarithmic Differentiation
- Differentiating the natural logarithm of both sides of an equation, so that products and powers become sums and multiples.
Logarithmic and Exponential Derivatives FAQ
What is the difference between the derivative of x squared and the derivative of 2 to the power x?
The first has a constant exponent and variable base, so the power rule gives 2x. The second has a constant base and variable exponent, so the exponential rule gives 2 to the power x times ln 2. Confusing the two is common; ask which part is constant before choosing a rule.
Why is e the natural base for calculus?
Because e to the power x is the one exponential function whose derivative is itself with no extra constant, which is the same as saying its graph has slope exactly one where it crosses the vertical axis. Every other base introduces a factor ln a, and every other logarithm a factor one over ln a.
When is logarithmic differentiation worth using?
When a function is a product or quotient of three or more factors, or has powers and roots mixed together, and always when the variable appears in both the base and the exponent. Taking logs turns the structure into a sum that differentiates term by term.
Exam move
Before differentiating any logarithm, rewrite it with the log laws and differentiate the simplified form; compare the length of the working with the chain-rule route once, and the habit will stick. Keep a short list of functions with variable exponents and practise logarithmic differentiation on each.
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