Objective – Logarithms, Lesson: Derivative Techniques 5 Objective – Logarithms, Euler’s Number, & Differentiation, oh my!

1. What is a Logarithm? A logarithm can be defined as an exponent Consider the following logarithmic expression It represents the exponent that the base of 7 must be raised to, in order to get the value 49.

2. What is Euler’s Number? e is a real number constant that appears in some kinds of mathematics problems. Examples of such problems are those involving growth or decay (including compound interest), the statistical "bell curve," the shape of a hanging cable (or the Gateway Arch in St. Louis), some problems of probability, some counting problems, and even the study of the distribution of prime numbers. It appears in Stirling's Formula for approximating factorials. It also shows up in calculus quite often, wherever you are dealing with either logarithmic or exponential functions. There is also a connection between e and complex numbers, via Euler's Equation.

The number e was first studied by the Swiss mathematician Leonhard Euler in the 1720s, although its existence was more or less implied in the work of John Napier, the inventor of logarithms, in 1614. Euler was also the first to use the letter e for it in 1727 (the fact that it is the first letter of his surname is coincidental). As a result, sometimes e is called the Euler Number, the Eulerian Number, or Napier's Constant.

3. What is the Natural Logarithm (LN)? A logarithm with a base of e 4. Properties of Logs

Formulas:

Basically, think of derivatives of “e” and “Ln” as chain rule problems.

Let’s try a few:

Proof: Let’s start with: LN both sides: Use log property: Simplify: What’s up with ex? How can it be the derivative of itself? Proof: Let’s start with: LN both sides: Use log property: Simplify:

Simplify: Derive both sides with respect to x: Multiply both sides by y: Simplify: Replace y with the original equation: Q.E.D.

Cool, isn’t it? why not also prove that: Proof: Let , so Derive both sides, with respect to x Solve for y’ Therefore: QED

EX. 7: Find

EX. 8: Find

EX. 9: Find If the mere sight of this problem makes you want to break stuff & cry don’t worry. You are not alone. But fear not, it is very workable with the use of the logarithmic properties. Now isn’t that special?

EX. 10: Find y’, if

EX. 11: Find

EX. 12: Find

Deriving the exponential function Remember: What about Stuff

Ex. 13: Differentiate

HW: Log Differentiation Worksheet