Glossary

1.3 Exponential and Logarithmic Functions

3 min read•august 7, 2024

Exponential and logarithmic functions are powerful tools in calculus, modeling real-world phenomena like population growth and radioactive decay. These functions are inverses of each other, with exponential functions growing or decaying rapidly and logarithmic functions increasing more slowly.

Understanding these functions is crucial for solving complex equations and analyzing data in fields like finance, science, and engineering. They provide a foundation for more advanced calculus concepts and are essential for modeling many natural and economic processes.

Exponential Functions

Definition and Characteristics

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  • takes the form f(x)=bxf(x) = b^x, where bb is a positive real number not equal to 1 called the and xx is any real number
  • is a special case where the base is the mathematical constant ee (approximately 2.71828), expressed as f(x)=[ex](https://www.fiveableKeyTerm:ex)f(x) = [e^x](https://www.fiveableKeyTerm:e^x)
  • Exponential functions are always positive, never zero or negative, as the base is raised to a power
  • Exponential functions have a horizontal at y=0y = 0 when the base is between 0 and 1, and no horizontal asymptote when the base is greater than 1

Growth and Decay Models

  • occurs when the rate of change is proportional to the current value, resulting in the function increasing at an increasingly rapid rate (b>1b > 1)
  • happens when the rate of change is proportional to the current value, causing the function to decrease at a decreasing rate (0<b<10 < b < 1)
  • Real-world applications of exponential growth include population growth, , and viral spread (COVID-19 pandemic)
  • Exponential decay models can represent radioactive decay, cooling of objects, and drug elimination from the body ( of medication)

Solving Exponential Equations

  • Exponential equations involve the variable in the exponent, such as 2x=82^x = 8
  • To solve exponential equations with the same base on both sides, set the exponents equal and solve for the variable (2x=232^x = 2^3 implies x=3x = 3)
  • When the bases are different, use logarithms to rewrite the equation and solve for the variable (2x=82^x = 8 can be rewritten as log⁡28=x\log_2 8 = x, giving x=3x = 3)
  • Applications of exponential equations include determining doubling time, half-life, and carbon dating (estimating the age of organic materials)

Logarithmic Functions

Definition and Properties

  • Logarithm is the inverse operation of exponentiation, denoted as log⁡bx=y\log_b x = y if and only if by=xb^y = x, where bb is the base, xx is the argument, and yy is the logarithm
  • (ln⁡\ln) is the logarithm with base ee, the mathematical constant approximately equal to 2.71828
  • include product rule (log⁡b(MN)=log⁡bM+log⁡bN\log_b (MN) = \log_b M + \log_b N), quotient rule (log⁡b(M/N)=log⁡bM−log⁡bN\log_b (M/N) = \log_b M - \log_b N), and (log⁡b(Mn)=nlog⁡bM\log_b (M^n) = n \log_b M)
  • Logarithms can be used to simplify calculations involving large numbers or powers (pH scale, Richter scale for earthquake magnitudes)

Solving Logarithmic Equations

  • Logarithmic equations have the variable inside a logarithm, such as log⁡2x=3\log_2 x = 3
  • To solve logarithmic equations, rewrite the equation in exponential form and solve for the variable (log⁡2x=3\log_2 x = 3 becomes 23=x2^3 = x, giving x=8x = 8)
  • When multiple logarithms are present, use the to combine or separate terms before solving (log⁡3x+log⁡3(x−1)=2\log_3 x + \log_3 (x-1) = 2 can be rewritten as log⁡3(x(x−1))=2\log_3 (x(x-1)) = 2)
  • Applications of logarithmic equations include determining the time required for an investment to reach a specific value or the intensity of sound (decibel scale)

Logarithmic and Exponential Relationships

Change of Base Formula

  • allows converting logarithms from one base to another: log⁡bx=log⁡axlog⁡ab\log_b x = \frac{\log_a x}{\log_a b}, where aa is any positive real number not equal to 1
  • Common use is to rewrite logarithms with base 10 or base ee in terms of natural logarithms or common logarithms (log⁡2x=ln⁡xln⁡2\log_2 x = \frac{\ln x}{\ln 2})
  • The change of base formula is derived from the properties of logarithms and exponential functions
  • Applying the change of base formula simplifies calculations when using logarithmic or exponential models (Fibonacci sequence, golden ratio)

Key Terms to Review (27)

Asymptote: An asymptote is a line that a graph approaches but never actually touches or intersects. This concept is crucial for understanding the behavior of various functions, especially as they extend towards infinity or exhibit discontinuities. Asymptotes can be horizontal, vertical, or oblique, each revealing different characteristics of the function's graph and its limits.
Base: In mathematics, particularly in exponential and logarithmic functions, a base refers to the number that is raised to a power. The base is crucial because it determines the growth rate of the function when it is raised to different exponents or used in logarithmic calculations. Understanding the concept of base helps in analyzing how exponential functions behave and how logarithms are inversely related to these functions.
Change of Base Formula: The change of base formula is a mathematical rule that allows you to convert logarithms from one base to another, making it easier to compute and analyze logarithmic expressions. This formula is especially useful when dealing with logarithms that are not in the form of a standard calculator base, such as base 10 or base e, by transforming them into a more manageable form. Understanding this formula is essential for working with exponential and logarithmic functions, as well as for deriving their properties and applications.
Compound Interest: Compound interest is the interest calculated on the initial principal and also on the accumulated interest from previous periods. This means that over time, the interest earns interest, leading to exponential growth of the investment or loan. It highlights the importance of time in finance, demonstrating how money can grow significantly when left to accumulate for longer periods.
Derivative of e^x: The derivative of e^x is the rate at which the function e^x changes with respect to x, and it is unique because the derivative is equal to the original function itself. This property makes the exponential function particularly important in calculus and mathematical analysis. Understanding this derivative lays the groundwork for exploring more complex topics in differential calculus and its applications.
Domain: In mathematics, the domain refers to the complete set of possible values of the independent variable(s) for which a function is defined. It essentially outlines the input values that can be plugged into a function without leading to any undefined or nonsensical results. Understanding the domain is crucial because it helps identify the valid inputs, ensuring that the outputs of the function make sense within its context.
E^x: The expression e^x represents the exponential function where the base is the mathematical constant e, approximately equal to 2.71828. This function is significant in many areas of mathematics and science, particularly in modeling growth processes, decay, and complex phenomena where rates of change are proportional to their current value. The exponential function e^x is unique because its rate of growth is proportional to its value at any point, making it a key function in calculus and analytic geometry.
Exponential Decay: Exponential decay refers to a process where a quantity decreases at a rate proportional to its current value. This concept is often represented mathematically as an exponential function, showing how the quantity diminishes over time. It commonly appears in real-world scenarios such as radioactive decay, population decline, and the cooling of objects, linking closely with exponential functions and logarithmic functions for understanding the behavior of decaying processes.
Exponential Decay Model: An exponential decay model describes a process where a quantity decreases at a rate proportional to its current value, leading to a rapid decrease initially that slows over time. This model is crucial for understanding phenomena such as radioactive decay, population decline, and the cooling of objects, as it helps predict how a quantity will diminish over time. The mathematical representation of this model typically involves an exponential function in the form of $$y = ae^{-kt}$$, where 'a' is the initial amount, 'k' is the decay constant, and 't' represents time.
Exponential Function: An exponential function is a mathematical expression in the form $$f(x) = a \cdot b^{x}$$, where 'a' is a constant, 'b' is a positive real number, and 'x' is the variable exponent. This type of function exhibits rapid growth or decay and is fundamental in modeling real-world phenomena such as population growth, radioactive decay, and compound interest. Exponential functions are closely related to logarithmic functions, allowing for conversions between exponential and logarithmic forms.
Exponential Growth: Exponential growth refers to the increase in a quantity where the rate of growth is proportional to its current value, leading to rapid escalation over time. This type of growth is characterized by the mathematical expression $$y = a(1 + r)^t$$, where 'a' represents the initial amount, 'r' is the growth rate, and 't' is time. As time progresses, exponential growth can lead to significantly large numbers in a relatively short period, making it a key concept in various fields such as biology, finance, and population studies.
Graph of y = e^x: The graph of y = e^x represents an exponential function where the base e (approximately equal to 2.718) is raised to the power of x. This graph is essential in understanding exponential growth, displaying unique characteristics such as an increasing trend, a horizontal asymptote at y = 0, and passing through the point (0, 1). Recognizing these features helps in analyzing real-world applications such as population growth and compound interest.
Graph of y = log(x): The graph of y = log(x) represents the logarithmic function, where y is the logarithm of x to a specified base, typically base 10 or base e (natural logarithm). This graph showcases how logarithmic functions behave, illustrating key features like their asymptotic nature, domain, and range, and provides a visual representation of exponential growth in reverse. Understanding this graph is crucial for comprehending the relationships between exponential and logarithmic functions.
Growth rate: The growth rate is a measure of how quickly a quantity increases over a specific period of time, often expressed as a percentage. It is particularly important in understanding the behavior of functions, especially exponential functions, where small changes can lead to significant differences in outcomes. Growth rates help to describe phenomena such as population increase, financial returns, or the spread of diseases, illustrating how quickly or slowly these changes occur.
Half-Life: Half-life is the time required for half of the quantity of a substance to undergo decay or transformation, typically used in the context of radioactive decay and exponential decay processes. This concept is crucial for understanding how quickly substances diminish over time, making it a key feature in modeling various natural phenomena. It also connects to exponential functions, as the amount remaining can be represented with exponential decay equations.
Horizontal Shift: A horizontal shift refers to the transformation of a function that moves it left or right on the Cartesian plane. This shift occurs when a constant is added or subtracted inside the function's argument, altering the input values required to achieve the same output. In the context of exponential and logarithmic functions, understanding horizontal shifts helps in graphing these functions accurately and interpreting their behavior, particularly how they affect growth and decay rates.
Integral of ln(x): The integral of ln(x) is the process of finding the antiderivative of the natural logarithmic function, which results in a function that describes the accumulation of the area under the curve of ln(x). This integral can be computed using integration by parts, highlighting the relationship between exponential and logarithmic functions, as well as the properties of integrals in calculus.
Log_b(x): The expression log_b(x) represents the logarithm of x to the base b, which answers the question: 'To what power must b be raised to obtain x?'. This concept connects exponential functions and logarithmic functions, showcasing how they are inverses of each other. Understanding log_b(x) is crucial for solving equations involving exponential growth and decay, as well as for working with scales that require logarithmic measures, such as pH levels or decibels.
Logarithmic equation: A logarithmic equation is an equation that involves a logarithm of an expression, typically written in the form $$ ext{log}_b(x) = y$$, where $$b$$ is the base, $$x$$ is the argument, and $$y$$ is the output. These equations are crucial for solving problems involving exponential growth and decay, as they allow us to find the exponent when given a specific value of the base raised to that exponent. Logarithmic equations often require properties of logarithms to be solved effectively.
Logarithmic Function: A logarithmic function is the inverse of an exponential function, expressing the relationship between a number and its exponent. Specifically, if $y = b^x$, then the logarithmic function is defined as $x = ext{log}_b(y)$, where $b$ is the base of the logarithm. This relationship highlights how logarithmic functions help to solve for unknown exponents and are essential for modeling various real-world situations involving growth and decay.
Logarithmic Properties: Logarithmic properties are the fundamental rules that govern the manipulation and evaluation of logarithms, which are the inverses of exponential functions. These properties help simplify complex logarithmic expressions and equations by establishing relationships between logarithms of different bases, products, quotients, and powers. Understanding these properties is crucial for solving equations involving logarithms and for connecting them to their corresponding exponential forms.
Natural Exponential Function: The natural exponential function is defined as the function $$f(x) = e^x$$, where $$e$$ is an irrational constant approximately equal to 2.71828. This function is significant because it serves as the base for natural logarithms and is widely used in calculus, especially in growth and decay problems. Its unique property is that the derivative of the function is equal to the function itself, making it a key player in differential equations and many areas of applied mathematics.
Natural Logarithm: The natural logarithm, denoted as $$ ext{ln}(x)$$, is the logarithm to the base of the mathematical constant $$e$$ (approximately 2.71828). It is a critical function in mathematics, particularly in calculus and exponential growth or decay scenarios. The natural logarithm is useful for solving equations involving exponential functions and plays a significant role in calculus due to its unique properties, such as its derivative being equal to the reciprocal of its argument.
Power Rule: The power rule is a basic principle in calculus that provides a quick way to differentiate functions of the form $$f(x) = x^n$$, where $$n$$ is any real number. This rule states that the derivative of such a function is given by multiplying the coefficient by the exponent and then reducing the exponent by one, resulting in $$f'(x) = n \cdot x^{n-1}$$. This concept is foundational in understanding how to work with various types of functions, especially in the context of differentiation and integration.
Properties of Logarithms: The properties of logarithms are fundamental rules that govern the manipulation of logarithmic expressions. These properties help simplify complex logarithmic equations, making it easier to solve for unknown variables. They include rules for combining logarithms, changing bases, and transforming multiplication and division into addition and subtraction within logarithmic contexts.
Range: The range of a function refers to the set of all possible output values (or dependent variables) that can result from plugging in values from the domain (or independent variables) into the function. Understanding the range helps in visualizing how a function behaves and where its values can lie, providing insight into important characteristics such as maximum and minimum points, which are crucial when analyzing algebraic and exponential/logarithmic functions.
Vertical Stretch: A vertical stretch occurs when a function is multiplied by a factor greater than 1, causing its graph to elongate vertically away from the x-axis. This transformation affects the output values of the function, making them larger without altering the x-values. The result is that points on the graph are moved further from the x-axis, changing the overall shape of the function and potentially altering its intercepts and asymptotic behavior.
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