Introduction to Logarithms: Definitions, Conversions, and Properties

Logarithms Introduction

Definition and Inverse Relationship

Logarithms are mathematical operations that serve as the inverse of exponentiation. They answer the question: "To what exponent must a base be raised to produce a given number?" Logarithms are essential in solving equations involving exponents and appear frequently in Precalculus and higher mathematics.

• Inverse Operation: The logarithm is the inverse of exponentiation.

• Base: The base of a logarithm is the number that is repeatedly multiplied in exponentiation.

• Logarithmic Equation: If $a^x = b$, then $x = ext{log}_a b$.

Converting Between Exponential and Logarithmic Forms

It is important to be able to convert between exponential and logarithmic forms for solving equations and understanding their properties.

• Exponential Form: $a^x = b$

• Logarithmic Form: $x = ext{log}_a b$

• Example: $3^4 = 81$ can be written as $ext{log}_3 81 = 4$

Exponential Form	Logarithmic Form
$3^4 = 81$	$ext{log}_3 81 = 4$
$x^a = b$	$a = ext{log}_x b$
$e^x = 64$	$x = ext{log}_e 64$

Examples of Conversion

• Convert $10^4 = 10,000$ to logarithmic form: $ext{log}_{10} 10,000 = 4$

• Convert $ext{log}_{5400} x = 3$ to exponential form: $x = 5400^3$

The Natural Logarithm

Definition and Properties

The natural logarithm uses the base $e$ (Euler's number, approximately 2.718). It is denoted as $ext{ln} x$ and is commonly used in calculus and higher mathematics.

• Common Logarithm: $ext{log}_{10} x$

• Natural Logarithm: $ext{log}_e x = ext{ln} x$

• Exponential Form: $e^x = a$

• Logarithmic Form: $x = ext{ln} a$

• Example: $e^x = 17$ is equivalent to $x = ext{ln} 17$

Properties of Logarithms

Fundamental Properties

Logarithms have several key properties that simplify calculations and allow for the evaluation of complex expressions.

Name	Property	Example	Description
Inverse Property	$ext{log}_a a^x = x$	$ext{log}_2 2^3 = 3$	Logarithm and exponent with same base cancel
Identity Property	$ext{log}_a 1 = 0$	$ext{log}_5 1 = 0$	Any log of 1 is zero
Log of Base	$ext{log}_a a = 1$	$ext{log}_7 7 = 1$	Log of its base equals 1

Examples of Evaluating Logarithms

• Evaluate $ext{log}_2 rac{1}{2}$: $-1$

• Evaluate $ext{log}_3 1$: $0$

• Evaluate $ext{log}_{10} 10$: $1$

• Evaluate $ext{log}_5 25$: $2$

Practice Problems

Converting Logarithmic and Exponential Forms

• Convert $ext{log}_2 x = 1$ to exponential form: $x = 2^1$

• Convert $x = ext{log}_{10} 10$ to exponential form: $10^x = 10$

• Convert $3^x = 7$ to logarithmic form: $x = ext{log}_3 7$

• Convert $4^x = 2$ to logarithmic form: $x = ext{log}_4 2$

Evaluating Logarithms

• Evaluate $ext{log}_2 3.5$: $1.79$

• Evaluate $rac{2}{3} ext{log}_5 8$: $0.52$

• Evaluate $ext{log}_3 rac{1}{81}$: $-4$

Additional info: These notes cover the essential introduction to logarithms, including definitions, conversions, properties, and practice problems, suitable for Precalculus students.