## Chapter 5

# Exponents and Logarithms

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By Boundless

By Boundless

Boundless Algebra

Algebra

by Boundless

Section 1

Inverse Functions

Inverses

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

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.

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.

Restricting Domains

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

Section 2

Graphing Exponential Functions

Basics of Graphing Exponential Functions

The exponential function $y=a\cdot b^{x}$ is a function that will remain proportional to its original value when it grows or decays.

Problem-Solving

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

e

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

Graphs of Exponential Functions, Base e

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

Section 3

Graphing Logarithmic Functions

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.

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.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.

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.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.

Section 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.Section 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: $P(t)=\frac{c}{1+a\cdot e^{-bt}}$ .

Exponential Decay

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

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