# Exponents and Logarithms

## 5.1 Inverse Functions

**5**.1.1

#### Inverses

Logarithm reverses exponentiation. The complex logarithm is the inverse function of the exponential function applied to complex numbers.

**5**.1.2

#### One-to-One Functions

A one-to-one function, also called an injective function, never maps distinct elements of its domain to the same element of its codomain.

**5**.1.3

#### Finding Formulas for Inverses

To find the inverse function, switch the x and y values, and then solve for y.

#### Composition and Composite Functions

A composite function represents, in one function, the results of an entire chain of dependent functions.

**5**.1.5

#### Restricting Domains

Domain restriction is important for inverse functions of exponents and logarithms because sometimes we need to find an unique inverse.

## 5.2 Graphing Exponential Functions

**5**.2.1

#### Basics of Graphing Exponential Functions

The exponential function

**5**.2.2

#### Problem-Solving

Graphically solving problems with exponential functions allows visualization of sometimes complicated interrelationships.

**5**.2.3

#### e

The number e is an important mathematical constant, approximately equal to 2.71828, that is the base of the natural logarithm, ln(x).

**5**.2.4

#### Graphs of Exponential Functions, Base e

The function $f(x)=e^{x}$ is a basic exponential function with some very interesting properties.

## 5.3 Graphing Logarithmic Functions

**5**.3.1

#### Logarithmic Functions

The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number.

**5**.3.2

#### Special Logarithms

Any number can be used as the base of a logarithm but certain bases (10, *e*, and 2) have more widespread applications than others.

**5**.3.3

#### Converting between Exponential and Logarithmic Equations

Logarithmic and exponential forms are closely related, and an equation in either form can be freely converted into the other.

**5**.3.4

#### Natural Logarithms

The natural logarithm is the logarithm to the base *e*, where *e* is an irrational and transcendental constant approximately equal to 2.718281828.

**5**.3.5

#### Changing Logarithmic Bases

The base of a logarithm can be converted to another value through a simple, one-step process.

#### Graphs of Logarithmic Functions

Logarithms can be graphed manually or electronically with points generally determined via a calculator or table.

#### Solving Problems with Logarithmic Graphs

Some functions with rapidly changing shape are best plotted on a scale that increases exponentially; such scales make up logarithmic graphs.

## 5.4 Properties of Logarithmic Functions

#### Logarithms of Products

A useful property of logarithms states that the sum of two logarithms of factors is equal to the logarithm of the factors' product.

#### Logarithms of Powers

A simplifying principle of logarithms is that the logarithm of the p-th power of a number is p times the logarithm of the number.

#### Logarithms of Quotients

The logarithm of the ratio or quotient of two numbers is the difference of the logarithms and can be proven using the first law of exponents.

#### Solving General Problems with Logarithms and Exponents

Logarithms are useful for solving equations that require an exponential term, like population growth.

#### Simplifying Expressions of the Form log_a a^x and a(log_a x)

The expressions log_{a}a^{x} and a^{log}_{a}^{x} can be simplified to *x*, a shortcut in complex equations.

## 5.5 Growth and Decay; Compound Interest

#### Population Growth

Population size can fluctuate positively or negatively, and growth is capable of being modeled by an exponential function.

#### Interest Compounded Continuously

Compound interest is accrued when interest is earned not only on principal, but on previously accrued interest: it is interest on interest.

#### Limited Growth

Exponential growth may dampen approaching a certain value, modeled with the logistic growth model:

#### Exponential Decay

Just as a variable can exponentially increase as a function of another, it is possible for a variable to exponentially decrease.