Algebra
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Boundless Algebra
Exponents, Logarithms, and Inverse Functions
Introduction to Exponents and Logarithms
Algebra Textbooks Boundless Algebra Exponents, Logarithms, and Inverse Functions Introduction to Exponents and Logarithms
Algebra Textbooks Boundless Algebra Exponents, Logarithms, and Inverse Functions
Algebra Textbooks Boundless Algebra
Algebra Textbooks
Algebra
Concept Version 10
Created by Boundless

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.

Learning Objective

  • Identify the parts of a logarithmic function and their characteristics


Key Points

    • The inverse of the logarithmic operation is exponentiation.
    • The logarithm is commonly used in many fields: that with base 222 in computer science, that with base eee in pure mathematics and financial mathematics, and that with base 101010 in natural science and engineering.

Terms

  • exponentiation

    The process of calculating a power by multiplying together a number of equal factors, where the exponent specifies the number of factors to multiply.

  • exponent

    The power to which a number, symbol, or expression is to be raised. For example, the 333 in x3x^3x​3​​.

  • logarithm

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


Full Text

In its simplest form, a logarithm is an exponent. Taking the logarithm of a number, one finds the exponent to which a certain value, known as a base, is raised to produce that number once more.

Logarithms have the following structure: logb(x)=clog{_b}(x)=clog​b​​(x)=c where bbb is known as the base, ccc is the exponent to which the base is raised to afford xxx. The base b>0b>0b>0.

Note that $​​log{_b}x=c$ is not defined for c<0c<0c<0. This is because the base bbb is positive and raising a positive number to any power will yield a non-negative number.

Commonly Used Bases

A logarithm with a base of 101010 is called a common logarithm and is denoted simply as logxlogxlogx. The common log is used often in science and engineering.

A logarithm with a base of eee is called a natural logarithm and is denoted lnxlnxlnx. The irrational number  e≈2.718e\approx 2.718 e≈2.718 arises naturally in financial mathematics, in computations having to do with compound interest and annuities.

A logarithm with a base of 222 is called a binary logarithm. While it has no special notation, it is often used in computer science.

The Exponential and Logarithmic Forms of an Equation

Logarithmic equations can be written as exponential equations and vice versa. The logarithmic equation logb(x)=clog_b(x)=clog​b​​(x)=c corresponds to the exponential equation bc=xb^{c}=xb​c​​=x.

As an example, the logarithmic equation log216=4log{_2}16=4log​2​​16=4 corresponds to the exponential equation 24=162^4=162​4​​=16.

Example 1: Solve for xxx in the equation log3(243)=xlog{_3}(243)=xlog​3​​(243)=x. 

Here we are looking for the exponent to which 333 is raised to yield 243243243. 

It might be more familiar if we convert the equation to exponential form giving us:

$3^x=243 \\ 3^5 =243$ 

Thus, log3(243)=5log{_3}(243)=5log​3​​(243)=5.

The explanation of the previous example reveals the inverse of the logarithmic operation: exponentiation. Starting with 243243243, if we take its logarithm with base 333, then raise 333 to the logarithm, we will once again arrive at 243243243.

Trivial Logarithmic Identities

logb1=0log_{b}1=0 log​b​​1=0 as  b0=1b^0=1b​0​​=1 for b≠0b\neq 0 b≠0. Note that 00≠10^0\neq 1 0​0​​≠1. Rather, 000^00​0​​ is called an indeterminate form.

logbb=1log{_b}b=1log​b​​b=1 as b1=bb^1=bb​1​​=b

logb0=undefinedlog{_b}0=undefinedlog​b​​0=undefined, as there is no number x such that bx=0b^x=0b​x​​=0

The following two logarithmic identities can be verified by converting the logarithmic equation into an exponential equation as follows:

blogb(x)=xb^{log_{b}(x)}=x b​log​b​​(x)​​=x 

Converting this to a logarithmic equation yields: logb(x)=logb(x)log_{b}(x)=log_{b}(x)log​b​​(x)=log​b​​(x)

Converting logb(bx)=xlog_{b}(b^x)=xlog​b​​(b​x​​)=x to an exponential equation yields bx=bxb^x=b^xb​x​​=b​x​​

Applications of Logarithms

Historically, logarithms were invented by John Napier as a way of doing lengthy arithmetic calculations prior to the invention of the modern day calculator. 

More recently, logarithms are most commonly used to simplify complex calculations that involve high-level exponents. In chemistry, for example, pH and pKa are used to simplify concentrations and dissociation constants, respectively, of high exponential value. The purpose is to bring wide-ranging values into a more manageable scope. A dissociation constant may be smaller than 101010^{10}10​10​​, or higher than 10−5010^{-50}10​−50​​. Taking the logarithm of each brings the values into a more comprehensible scope (101010 to −50-50−50) .

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