Glossary

6.3 Logarithmic Functions

3 min read•june 24, 2024

Logarithmic functions are powerful tools that help us understand exponential relationships. They're the flip side of exponents, letting us solve complex problems in fields like science and finance.

These functions have unique properties that make them incredibly useful. From measuring earthquakes to calculating compound interest, logarithms simplify complex calculations and help us make sense of vast ranges of data.

Logarithmic Functions

Logarithmic and exponential form conversion

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  • log⁡b(x)=y\log_b(x) = y equivalent to by=xb^y = x
    • bb represents the , xx the , and yy the or
  • Convert logarithmic to exponential form by using the base as the 's base, the logarithm as the exponent, and the argument as the result (23=82^3 = 8)
  • Convert exponential to logarithmic form by using the exponent's base as the logarithm's base, the result as the argument, and the exponent as the logarithm (log⁡2(8)=3\log_2(8) = 3)
    • This conversion demonstrates the relationship between logarithmic and exponential functions

Evaluation of logarithms with bases

  • log⁡b(x)=log⁡a(x)log⁡a(b)\log_b(x) = \frac{\log_a(x)}{\log_a(b)}, where aa represents any base
    • Evaluates logarithms with any base using a calculator with a different base (usually base 10 or [e](https://www.fiveableKeyTerm:e)[e](https://www.fiveableKeyTerm:e))
  • Common bases include base 10 () log⁡(x)\log(x) or log⁡10(x)\log_{10}(x) and base ee () ln⁡(x)\ln(x) or log⁡e(x)\log_e(x), where e≈2.71828e \approx 2.71828
  • Logarithms can only be evaluated for positive arguments (x>0x > 0)

Domain, Range, and Asymptotes of Logarithmic Functions

  • Domain of logarithmic functions: all positive real numbers (x > 0)
  • Range of logarithmic functions: all real numbers
  • Vertical : x = 0, as the function approaches negative infinity as x approaches 0 from the right

Applications of Logarithms

Common logarithms in real-world applications

  • measures the magnitude of an earthquake
    • Richter magnitude calculated by log⁡(Seismic wave amplitudeStandard amplitude)\log(\frac{\text{Seismic wave amplitude}}{\text{Standard amplitude}})
    • An increase of 1 on the Richter scale corresponds to a tenfold increase in seismic wave amplitude
  • measures the intensity of sound
    • Decibels (dB) calculated by 10log⁡(Sound intensityReference intensity)10 \log(\frac{\text{Sound intensity}}{\text{Reference intensity}})
    • An increase of 10 dB corresponds to a tenfold increase in sound intensity
  • measures the acidity or alkalinity of a solution
    • pH calculated by −log⁡([H+])-\log([\text{H}^+]), where [H+][\text{H}^+] represents the concentration of hydrogen ions in moles per liter
    • A decrease of 1 pH unit corresponds to a tenfold increase in hydrogen ion concentration
  • These applications demonstrate the use of logarithmic scales in various scientific fields

Natural logarithms for growth and decay

  • modeled by A(t)=A0ektA(t) = A_0e^{kt}, where A0A_0 represents the initial amount, kk the growth rate, and tt time
    • calculated by td=ln⁡(2)kt_d = \frac{\ln(2)}{k}
  • modeled by A(t)=A0e−ktA(t) = A_0e^{-kt}, where A0A_0 represents the initial amount, kk the decay rate, and tt time
    • calculated by t1/2=ln⁡(2)kt_{1/2} = \frac{\ln(2)}{k}
  • determines the age of organic materials based on the decay of carbon-14
    • Remaining carbon-14 calculated by A(t)=A0e−ktA(t) = A_0e^{-kt}, where k=ln⁡(2)5730k = \frac{\ln(2)}{5730} (half-life of carbon-14 is 5730 years)

Solving equations with logarithmic properties

  • log⁡b(MN)=log⁡b(M)+log⁡b(N)\log_b(MN) = \log_b(M) + \log_b(N)
  • log⁡b(MN)=log⁡b(M)−log⁡b(N)\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)
  • log⁡b(Mn)=nlog⁡b(M)\log_b(M^n) = n\log_b(M)
  • Steps to solve logarithmic equations:
    1. Isolate the logarithm on one side of the equation
    2. Apply the appropriate logarithmic properties to simplify the equation
    3. Convert the equation to exponential form
    4. Solve the resulting

Key Terms to Review (44)

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.
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.
Asymptote: An asymptote is a line or curve that a graph approaches but never touches. It represents the limit of a function's behavior as the input variable approaches a particular value. Asymptotes are an important concept in various mathematical topics, including rational expressions, functions, rational functions, exponential functions, logarithmic functions, and exponential and logarithmic models.
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.
Carbon Dating: Carbon dating is a radiometric dating method used to determine the age of organic materials by measuring the amount of radioactive carbon-14 (14C) they contain. It is a crucial technique in fields such as archaeology, geology, and paleontology, allowing scientists to establish the age of archaeological artifacts, fossils, and other organic remains.
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.
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 $c \neq 1$.
Common logarithm: A common logarithm is a logarithm with base 10, often written as $\log_{10}(x)$ or simply $\log(x)$. It is commonly used in scientific calculations and when dealing with exponential growth or decay.
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.
Decibel Scale: The decibel scale is a logarithmic unit used to measure the intensity or level of various quantities, such as sound, power, and voltage. It is commonly used in the context of acoustics, electronics, and telecommunications to quantify the relative change in a measured value.
Domain and range: Domain is the set of all possible input values for a function, while range is the set of all possible output values. Together, they describe the scope of a function's operation.
Domain and Range: The domain of a function refers to the set of input values that the function is defined for, while the range of a function refers to the set of output values that the function can produce. Understanding domain and range is crucial in analyzing the behavior and characteristics of various functions, including inverses, exponential, and logarithmic functions.
Doubling time: Doubling time is the period it takes for a quantity to double in size or value at a constant growth rate. It is commonly used in exponential growth models.
Doubling Time: Doubling time is the amount of time it takes for a quantity to double in value. It is a crucial concept in the study of exponential growth and decay, and is closely tied to the understanding of exponential functions, their graphs, logarithmic functions, and their applications in various models.
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.
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.
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 Decay: Exponential decay is a mathematical model that describes the gradual reduction or diminishment of a quantity over time. It is characterized by an initial value that decreases by a constant proportion during each successive time interval, resulting in an exponential decrease. This concept is fundamental to understanding various phenomena in fields such as physics, chemistry, biology, and finance.
Exponential equation: An exponential equation is an equation in which the variables appear as exponents. These equations often take the form $a^{x} = b$ where $a$ and $b$ are constants.
Exponential Equation: An exponential equation is a mathematical equation in which the unknown variable appears as the exponent of another quantity. These equations model situations where a quantity grows or decays at a constant rate over time, and are closely related to the behavior of exponential functions.
Exponential Form: Exponential form refers to the representation of a quantity or expression using an exponent. It is a concise way to express repeated multiplication, where the exponent indicates the number of times the base is multiplied by itself.
Exponential growth: Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value. This results in the function increasing rapidly over time.
Exponential Growth: Exponential growth is a pattern of change where a quantity increases at a rate proportional to its current value. This means the quantity grows by a consistent percentage over equal intervals of time, leading to rapid, accelerating growth. Exponential growth is a fundamental concept in mathematics and has applications across various fields, including biology, economics, and technology.
Half-life: Half-life is the time it takes for a radioactive or other substance to decay to half of its initial value. This concept is central to understanding exponential functions, their graphs, logarithmic functions, and how these models are applied to real-world situations involving growth and decay.
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 $f^{-1}(f(x)) = x$.
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.
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.
Logarithm: A logarithm is a mathematical function that describes the power to which a base number must be raised to get a certain value. It represents the exponent to which a base number must be raised to produce a given number. Logarithms are closely related to exponential functions and are essential in understanding topics such as logarithmic functions, graphs of logarithmic functions, exponential and logarithmic equations, and geometric sequences.
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.
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 Form: Logarithmic form is a way of expressing exponential relationships using logarithms. It allows for the representation of exponential functions in a linear manner, making them easier to analyze and work with mathematically.
Logarithmic Scale: A logarithmic scale is a way of representing numerical data where the spacing between values is proportional to their logarithmic values. This type of scale is commonly used to visualize data that spans a wide range of values, as it allows for the efficient display of both small and large numbers on the same axis.
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 $\ln(x)$.
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.
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.
PH Scale: The pH scale is a measure of the acidity or basicity of a solution, ranging from 0 to 14. It is a logarithmic scale that indicates the concentration of hydrogen ions (H+) in a solution, with lower values representing more acidic solutions and higher values representing more basic (alkaline) solutions.
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.
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.
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.
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, $\log_b(xy) = \log_b(x) + \log_b(y)$.
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.
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.
Richter Scale: The Richter scale is a logarithmic scale used to measure the magnitude of earthquakes. It was developed in 1935 by American seismologist Charles Richter and is a fundamental tool in understanding the strength and impact of seismic events.
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