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College Algebra
Unit 6 Overview: Exponential and Logarithmic Functions
6.1 Exponential Functions
6.2 Graphs of Exponential Functions
6.3 Logarithmic Functions
6.4 Graphs of Logarithmic Functions
6.5 Logarithmic Properties
6.6 Exponential and Logarithmic Equations
6.7 Exponential and Logarithmic Models
6.8 Fitting Exponential Models to Data

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6.5 Logarithmic Properties

Last Updated on June 24, 2024

Logarithmic properties are powerful tools for simplifying and solving complex equations. These rules help us manipulate expressions involving logs, making calculations easier and more efficient. Understanding these properties is crucial for tackling advanced algebraic problems.

From product and quotient rules to change-of-base formulas, logarithmic properties have wide-ranging applications. They're used in real-world scenarios like measuring earthquake intensity and sound levels. Mastering these concepts opens doors to solving intricate mathematical puzzles and understanding natural phenomena.

Logarithmic Properties

Rules for logarithmic simplification

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  • Product Rule: log⁡b(M⋅N)=log⁡b(M)+log⁡b(N)\log_b(M \cdot N) = \log_b(M) + \log_b(N) states that the logarithm of a product equals the sum of the logarithms of its factors
    • Example: log⁡2(6⋅8)=log⁡2(6)+log⁡2(8)\log_2(6 \cdot 8) = \log_2(6) + \log_2(8)
  • Quotient Rule: log⁡b(MN)=log⁡b(M)−log⁡b(N)\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N) indicates that the logarithm of a quotient equals the difference of the logarithms of the numerator and denominator
    • Example: log⁡5(255)=log⁡5(25)−log⁡5(5)\log_5(\frac{25}{5}) = \log_5(25) - \log_5(5)
  • Power Rule: log⁡b(Mn)=n⋅log⁡b(M)\log_b(M^n) = n \cdot \log_b(M) signifies that the logarithm of a number raised to a power equals the power multiplied by the logarithm of the number
    • Example: log⁡3(92)=2⋅log⁡3(9)\log_3(9^2) = 2 \cdot \log_3(9)
    • This rule demonstrates the relationship between logarithms and exponents

Expansion and condensation of logarithms

  • Expanding logarithmic expressions involves applying the product rule to split a logarithm of a product into a sum of logarithms, the quotient rule to split a logarithm of a quotient into a difference of logarithms, and the power rule to convert a logarithm of a power into a product of the power and the logarithm
    • Example: log⁡2(3x2y)=log⁡2(3)+2log⁡2(x)+log⁡2(y)\log_2(3x^2y) = \log_2(3) + 2\log_2(x) + \log_2(y)
  • Condensing logarithmic expressions entails combining logarithms with the same base using the product rule in reverse, the quotient rule in reverse, and simplifying logarithms with powers using the power rule in reverse
    • Example: log⁡4(2)+log⁡4(x)−3log⁡4(y)=log⁡4(2xy3)\log_4(2) + \log_4(x) - 3\log_4(y) = \log_4(\frac{2x}{y^3})

Change-of-base formula for logarithms

  • Change-of-base formula: log⁡b(x)=log⁡a(x)log⁡a(b)\log_b(x) = \frac{\log_a(x)}{\log_a(b)} allows for converting a logarithm with base bb to a logarithm with base aa
  • Steps to apply the change-of-base formula:
    1. Identify the logarithm with the base you want to change (log⁡b(x)\log_b(x))
    2. Choose a new base (aa) that is more convenient for calculation (base 10 or base ee)
    3. Apply the change-of-base formula using the new base (log⁡a(x)log⁡a(b)\frac{\log_a(x)}{\log_a(b)})
    4. Evaluate the resulting logarithms using a calculator or known values
  • Example: log⁡3(5)=log⁡(5)log⁡(3)≈1.465\log_3(5) = \frac{\log(5)}{\log(3)} \approx 1.465

Solving logarithmic equations

  • Isolating the logarithm using algebraic techniques to have the logarithmic term on one side of the equation
  • Applying the definition of logarithms: if log⁡b(x)=y\log_b(x) = y, then by=xb^y = x. After isolating the logarithm, rewrite the equation in exponential form
  • Solving the resulting exponential equation using properties of exponents to solve for the variable
  • Checking the solution by verifying that it satisfies the original logarithmic equation and ensuring that the solution is in the domain of the logarithmic function (argument must be positive)
  • Example: Solve log⁡2(3x−1)=4\log_2(3x-1) = 4
    1. Isolate the logarithm: log⁡2(3x−1)=4\log_2(3x-1) = 4
    2. Apply the definition of logarithms: 24=3x−12^4 = 3x-1
    3. Solve the exponential equation: 16=3x−116 = 3x-1, 17=3x17 = 3x, x=173x = \frac{17}{3}
    4. Check the solution: log⁡2(3(173)−1)=log⁡2(16)=4\log_2(3(\frac{17}{3})-1) = \log_2(16) = 4, and 173>0\frac{17}{3} > 0

Real-world applications of logarithms

  • Exponential growth and decay: logarithmic functions can model situations involving exponential growth or decay (population growth, radioactive decay, compound interest)
  • Richter scale for earthquake magnitudes: a logarithmic scale where each unit increase corresponds to a tenfold increase in the earthquake's magnitude
  • Decibel scale for sound intensity: a logarithmic scale where each 10-decibel increase corresponds to a tenfold increase in sound intensity
  • pH scale for acidity and alkalinity: a logarithmic scale where each unit decrease in pH corresponds to a tenfold increase in the concentration of hydrogen ions

Properties of logarithmic graphs

  • Domain: (0,∞)(0, \infty); the argument of a logarithm must be positive. Range: (−∞,∞)(-\infty, \infty); logarithms can take on any real value
  • Vertical asymptote: logarithmic functions have a vertical asymptote at x=0x = 0. As xx approaches 0 from the right, the logarithm approaches negative infinity
  • Increasing or decreasing: logarithmic functions are increasing if the base is greater than 1 (b>1b > 1) and decreasing if the base is between 0 and 1 (0<b<10 < b < 1)
  • Transformations:
    • Vertical shift: log⁡b(x)+k\log_b(x) + k shifts the graph up by kk units
    • Horizontal shift: log⁡b(x−h)\log_b(x - h) shifts the graph right by hh units
    • Vertical stretch or compression: a⋅log⁡b(x)a \cdot \log_b(x) stretches the graph vertically by a factor of ∣a∣|a| if ∣a∣>1|a| > 1 and compresses it if 0<∣a∣<10 < |a| < 1
    • Reflection: −log⁡b(x)-\log_b(x) reflects the graph across the xx-axis

Logarithms and inverse functions

  • Logarithmic functions are the inverse functions of exponential functions
  • The domain of a logarithmic function is the range of its corresponding exponential function, and vice versa
  • The graph of a logarithmic function is the reflection of its corresponding exponential function over the line y = x

Key Terms to Review (35)

Absolute maximum: The absolute maximum of a function is the highest value that the function attains over its entire domain. It represents the peak point on the graph of the function.
Change-of-base formula: The change-of-base formula is used to rewrite logarithms in terms of logs of another base, allowing for easier computation. It is commonly written as $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$ where $b$ and $c$ are positive real numbers and $c \neq 1$.
Domain: The domain of a function is the complete set of possible input values (x-values) that allow the function to work within its constraints. It specifies the range of x-values for which the function is defined.
Exponent: An exponent indicates how many times a number, known as the base, is multiplied by itself. It is written as a small number to the upper right of the base.
Inverse function: An inverse function reverses the operation of a given function. If $f(x)$ is a function, its inverse $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.
Logarithmic equation: A logarithmic equation is an equation that involves the logarithm of an expression containing a variable. Solving these equations often requires using properties of logarithms or converting them to exponential form.
Natural logarithm: The natural logarithm is the logarithm to the base $e$, where $e$ is an irrational and transcendental number approximately equal to 2.71828. It is commonly denoted as $\ln(x)$.
PH: pH is a measure of the acidity or alkalinity of a solution, defined as the negative logarithm (base 10) of the hydrogen ion concentration. It ranges from 0 to 14, with lower values being more acidic and higher values more basic.
Power rule for logarithms: The power rule for logarithms states that the logarithm of a number raised to an exponent is equal to the exponent times the logarithm of the number. Mathematically, $\log_b(a^c) = c \cdot \log_b(a)$ where $b$ is the base.
One-to-one: A one-to-one function is a function in which each element of the range is paired with exactly one element of the domain. This implies that no two different inputs produce the same output, ensuring the function passes the horizontal line test.
Quotient rule for logarithms: The quotient rule for logarithms states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator: $\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N$. It simplifies complex expressions involving division inside a logarithm.
Product rule for logarithms: The product rule for logarithms states that the logarithm of a product is equal to the sum of the logarithms of its factors. Mathematically, $\log_b(xy) = \log_b(x) + \log_b(y)$.
Antilogarithm: The antilogarithm is the inverse operation of the logarithm. It is the process of finding the original number or value when given its logarithm. The antilogarithm is used to undo the effects of a logarithmic transformation and retrieve the original quantity or value.
Argument: An argument is a set of statements or premises that are used to support or justify a particular conclusion or claim. It is a logical structure that connects various pieces of information to make a coherent and persuasive case.
Base: The base is a fundamental component in various mathematical concepts, serving as a reference point or starting value. It is a crucial element in understanding exponents, exponential functions, logarithmic functions, and geometric sequences, among other topics.
Change of Base Formula: The change of base formula is a mathematical expression that allows for the conversion of logarithms from one base to another. This formula is particularly important in the context of logarithmic functions, their graphs, and the properties and equations involving logarithms and exponentials.
Common Logarithm: The common logarithm, also known as the base-10 logarithm, is a logarithmic function that expresses the power to which a base of 10 must be raised to obtain a given number. It is a fundamental concept in mathematics, with applications in various fields, including college algebra.
Condensing Logarithms: Condensing logarithms is the process of simplifying logarithmic expressions by combining multiple logarithmic terms into a single logarithmic term. This technique is particularly useful in the context of logarithmic properties, as it allows for the efficient manipulation and simplification of complex logarithmic expressions.
Domain: The domain of a function refers to the set of input values for which the function is defined. It represents the range of values that the independent variable can take on, and it is the set of all possible values that can be plugged into the function to produce a meaningful output.
E: e, also known as Euler's number, is a fundamental mathematical constant that is the base of the natural logarithm. It is an irrational number that is approximately equal to 2.71828 and is widely used in mathematics, science, and engineering. The term 'e' is central to the understanding of exponential functions, logarithmic functions, and their properties, which are crucial concepts in college algebra.
Expanding Logarithms: Expanding logarithms is the process of rewriting a logarithmic expression by applying the properties of logarithms to break it down into simpler, more manageable components. This technique is crucial in the context of understanding and manipulating logarithmic functions, particularly in the topic of Logarithmic Properties.
Exponent: An exponent is a mathematical symbol that indicates the number of times a base number is multiplied by itself. It represents the power to which a number or variable is raised, and it is a fundamental concept in algebra, exponential functions, logarithmic functions, and other areas of mathematics.
Exponential Functions: An exponential function is a mathematical function in which the independent variable appears as an exponent. These functions model situations where a quantity grows or decays at a constant rate over time, and they are characterized by an initial value and a constant growth or decay factor.
Inverse Function: An inverse function is a function that reverses the operation of another function. It undoes the original function, mapping the output back to the original input. Inverse functions are crucial in understanding the relationships between different mathematical concepts, such as domain and range, composition of functions, transformations, and exponential and logarithmic functions.
Ln: The natural logarithm, denoted as ln, is a logarithmic function that describes the power to which a base of e (approximately 2.718) must be raised to get a certain value. It is a fundamental mathematical concept that is closely related to exponential functions and is essential in understanding logarithmic functions, their graphs, and their properties.
Logarithm Product Theorem: The Logarithm Product Theorem states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. This theorem is a fundamental property of logarithms that allows for simplifying and manipulating expressions involving multiplication of numbers or variables.
Logarithmic Equation: A logarithmic equation is an equation that involves one or more logarithmic functions. These equations are used to model and solve problems involving exponential growth or decay, as well as to find unknown values in situations where the relationship between variables is logarithmic in nature.
Logarithmic Identities: Logarithmic identities are mathematical relationships that describe the behavior of logarithmic functions. These identities provide a framework for manipulating and simplifying logarithmic expressions, which is essential in the context of 6.5 Logarithmic Properties.
Logarithmic Properties: Logarithmic properties are the mathematical rules that describe the behavior and relationships between logarithmic functions. These properties provide a framework for understanding and working with logarithms, which are essential in the study of exponential and logarithmic functions.
Log: A logarithm is the exponent to which a base must be raised to get a certain number. It is a mathematical function that describes the power to which a fixed number, called the base, must be raised to produce a given value.
Natural Logarithm: The natural logarithm, denoted as $\ln(x)$, is a logarithmic function that represents the power to which the base $e$ must be raised to get the value $x$. The natural logarithm is a fundamental concept that underpins various topics in college algebra, including logarithmic functions, their graphs, properties, and applications in solving exponential and logarithmic equations, as well as modeling real-world phenomena.
Power Rule: The power rule is a fundamental concept in calculus that describes how to differentiate functions raised to a power. It provides a straightforward method for finding the derivative of expressions involving exponents and powers.
Product Rule: The product rule is a fundamental concept in mathematics that describes the derivative of a product of two functions. It states that the derivative of a product is equal to the product of the derivative of the first function and the second function, plus the product of the first function and the derivative of the second function.
Quotient Rule: The quotient rule is a fundamental mathematical concept that describes how to differentiate the ratio or quotient of two functions. It is a crucial tool in the study of calculus and is applicable across various mathematical domains, including exponents, radicals, logarithmic functions, and more.
Range: In mathematics, the range refers to the set of all possible output values (dependent variable values) that a function can produce based on its input values (independent variable values). Understanding the range helps in analyzing how a function behaves and what values it can take, connecting it to various concepts like transformations, compositions, and types of functions.
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Glossary
6.6 Exponential and Logarithmic Equations