Exponents and logarithms

Exponents

Logarithms

Introduction

A logarithm is defined so that log_b(w)=z such that w=z^b. A logarithm has two parameters – a base. It value of the logarithm is you what you need to raise the base to to get the value of which the logarithm is taken.

Some notation and the definition of Euler's number

In mathematics ln(x) and log(x) are both taken to mean log_e(x), where e is an irrational mathematical constant called Euler's number and can be defined in different ways.

e = \lim_{n\to\infty}(1+\frac{1}{n})^n
e = \sum_{n=0}^{\infty}\frac{1}{n!}

The infinite decimal of e begins 2.718281828459045\dots.

Properties of logarithms

Much like the properties of powers there are also useful properties of exponents.

log_b(xy) = log_b(x) + log_b(y)
log_b(\frac{x}{y}) = log_b(x) - log_b(y)
log_b(x)^y = y log_b(x)

Converting products to sums with logarithms

It is often easier to compute the value of a summation than it is to find the value of a product. Using the property of logarithms log(xy)=log(x)+log(y) it is possible to rewrite a product as a sum. (This example is taken from Chapter 1 of Stephen J. Miller's The Probability Lifesaver (ISBN: 9780691149554))

\prod_{k=0}^n \frac{356-k}{356}

Taking the logarithm.

log(\prod_{k=0}^n \frac{356-k}{356}) = \sum_{k=0}^n log(\frac{356-k}{356})

Calculus of exponentials and logarithms

Derivatives

Derivative of e^x

Derivative of ln(x)

Derivative of a^x

Derivative of log_b(x)

Indefinite integrals

Infinite power series