## Chapter 5

# Exponents and Logarithms

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

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

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

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

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

The exponential function

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

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

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

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

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

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

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

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

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

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

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

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.

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.

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

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

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

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

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

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