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6.1 Exponential Functions

4 min read•june 24, 2024

Exponential functions model growth and decay in various real-world scenarios. They're essential for understanding , population dynamics, and . These functions have unique properties that set them apart from other mathematical models.

Exponential equations can be solved using , which are the inverse of exponential functions. Applications of and decay are widespread in science and finance, making them crucial tools for modeling and predicting change over time.

Exponential Functions

Exponential functions with various bases

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  • defined as f(x)=[bx](https://www.fiveableKeyTerm:bx)f(x) = [b^x](https://www.fiveableKeyTerm:b^x), where bb is a positive constant not equal to 1
    • b>1b > 1 represents (population growth)
    • 0<b<10 < b < 1 represents (radioactive decay)
  • f(x)=[e](https://www.fiveableKeyTerm:e)xf(x) = [e](https://www.fiveableKeyTerm:e)^x, where e≈2.71828e \approx 2.71828 is the mathematical constant
    • Frequently used in and calculus ()
  • Solving exponential equations involves isolating the exponential term on one side
    • Take the logarithm of both sides using the appropriate
    • Solve for the variable

Modeling with exponential equations

  • formula A=P(1+rn)ntA = P(1 + \frac{r}{n})^{nt} models financial growth
    • AA represents the final amount
    • PP represents the principal (initial investment)
    • rr represents the annual interest rate as a decimal (5% = 0.05)
    • nn represents the number of times interest is compounded per year (monthly = 12)
    • tt represents the time in years
  • Exponential growth in populations modeled by P(t)=P0ektP(t) = P_0e^{kt}
    • P(t)P(t) represents the population at time tt
    • P0P_0 represents the initial population
    • kk represents the (birth rate - death rate)
    • tt represents the time elapsed
  • Exponential decay in radioactive materials modeled by A(t)=A0e−λtA(t) = A_0e^{-\lambda t}
    • A(t)A(t) represents the amount of radioactive material at time tt
    • A0A_0 represents the initial amount
    • λ\lambda represents the (related to )
    • tt represents the time elapsed

Features of exponential graphs

  • Domain includes all real numbers
  • Range includes all positive real numbers
  • at y=0y = 0 (x-axis) as the graph approaches but never touches the axis
  • at (0,1)(0, 1) for all exponential functions
  • Increasing if b>1b > 1, decreasing if 0<b<10 < b < 1
  • Always , indicating increasing rate of change
  • shift and stretch the graph:
    • Vertical shift f(x)=bx+kf(x) = b^x + k moves the graph up or down
    • Horizontal shift f(x)=bx−hf(x) = b^{x-h} moves the graph left or right
    • Vertical stretch/compression f(x)=a⋅bxf(x) = a \cdot b^x changes the steepness
    • Reflection f(x)=b−xf(x) = b^{-x} flips the graph over the y-axis

Applications of exponential growth

  • modeled by N(t)=N0ertN(t) = N_0e^{rt}
    • N(t)N(t) represents the number of bacteria at time tt
    • N0N_0 represents the initial number of bacteria
    • rr represents the growth rate (dependent on environmental factors)
    • tt represents the time elapsed
  • Radioactive decay modeled by N(t)=N0e−λtN(t) = N_0e^{-\lambda t}
    • N(t)N(t) represents the number of radioactive atoms at time tt
    • N0N_0 represents the initial number of atoms
    • λ\lambda represents the decay constant (related to half-life)
    • tt represents the time elapsed
  • modeled by T(t)=Ta+(T0−Ta)e−ktT(t) = T_a + (T_0 - T_a)e^{-kt}
    • T(t)T(t) represents the temperature of the object at time tt
    • TaT_a represents the ambient temperature (surrounding environment)
    • T0T_0 represents the initial temperature of the object
    • kk represents the cooling constant (dependent on material properties)
    • tt represents the time elapsed

Exponential Functions in Context

Exponential functions with various bases

  • Continuous growth and decay modeled by A(t)=A0ertA(t) = A_0e^{rt}
    • Used when growth or decay occurs continuously (population growth, radioactive decay)
    • A(t)A(t) represents the value at time tt
    • A0A_0 represents the initial value
    • rr represents the growth rate (positive for growth, negative for decay)
    • tt represents the time elapsed
  • Half-life is the time it takes for a quantity to reduce to half its initial value
    • For exponential decay, half-life is calculated by t1/2=ln⁡(2)λt_{1/2} = \frac{\ln(2)}{\lambda}
    • λ\lambda represents the decay constant (related to the rate of decay)

Modeling with exponential equations

  • modeled by A=PertA = Pe^{rt}
    • Interest is compounded continuously, meaning an infinite number of times per year
    • AA represents the final amount
    • PP represents the principal (initial investment)
    • rr represents the annual interest rate as a decimal (5% = 0.05)
    • tt represents the time in years
  • is the time it takes for a quantity to double its initial value
    • For exponential growth, is calculated by tdouble=ln⁡(2)rt_{double} = \frac{\ln(2)}{r}
    • rr represents the growth rate (birth rate - death rate)

Logarithms and Inverse Functions

  • Logarithms are the of exponential functions
  • The logarithm base bb of xx is the to which bb must be raised to get xx
  • Common logarithm bases include:
    • Base 10 (common logarithm): log⁡10(x)\log_{10}(x) or simply log⁡(x)\log(x)
    • Base ee (natural logarithm): log⁡e(x)\log_e(x) or ln⁡(x)\ln(x)
  • Logarithmic properties are used to solve exponential equations
  • The domain of a logarithmic function is all positive real numbers
  • The range of a logarithmic function is all real numbers

Key Terms to Review (37)

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.
Annual percentage rate (APR): The Annual Percentage Rate (APR) is the yearly interest rate charged on borrowed money or earned through an investment, expressed as a percentage. It includes fees or additional costs associated with the transaction.
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.
B^x: The expression $b^x$ represents an exponential function, where $b$ is the base and $x$ is the exponent. This function describes a relationship where the output value grows exponentially as the input variable $x$ increases. The value of $b$ determines the rate of growth, with $b > 1$ resulting in an increasing function and $0 < b < 1$ resulting in a decreasing function.
Bacterial Growth: Bacterial growth refers to the increase in the number of bacteria through cell division and reproduction. It is a fundamental process that allows bacteria to thrive and colonize various environments, including the human body, soil, and water sources.
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 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 $c \neq 1$.
Compound interest: Compound interest is the interest calculated on the initial principal and also on the accumulated interest of previous periods. It is commonly used in finance and investments to calculate growth over time.
Compound Interest: Compound interest is the interest earned on interest, where the interest accrued on a principal amount is added to the original amount, and the total then earns interest in the next period. This concept is fundamental to understanding the growth of investments and loans over time.
Concave Up: Concave up is a term used to describe the shape of a curve on a graph, where the curve bends upward, forming a U-shape. This term is particularly relevant in the context of quadratic functions and exponential functions, as it describes the overall shape and behavior of these types of functions.
Continuous Growth: Continuous growth refers to a steady, uninterrupted increase in a quantity or variable over time. It is a fundamental concept in the study of exponential functions, where a quantity grows at a constant rate, leading to a continuous and compounding increase.
Continuously Compounded Interest: Continuously compounded interest is a method of calculating interest where the interest is compounded continuously over time, rather than at discrete intervals like daily, monthly, or annually. This results in a higher effective interest rate compared to simple interest or discrete compounding.
Decay Constant: The decay constant is a fundamental parameter that describes the rate of exponential decay in a radioactive or other time-dependent process. It represents the probability of a particle or system undergoing a specific type of decay or transformation per unit of time.
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 function: An exponential function is a mathematical expression in the form $f(x) = a \cdot b^x$, where $a$ is a constant, $b$ is the base greater than 0 and not equal to 1, and $x$ is the exponent. These functions model growth or decay processes.
Exponential Function: An exponential function is a mathematical function in which the independent variable appears as an exponent. These functions exhibit a characteristic curve that grows or decays at a rate proportional to the current value, leading to rapid changes in output as the input increases.
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.
Growth Rate: The growth rate refers to the rate of change in a quantity over time. It is a measure of how quickly a quantity is increasing or decreasing, and is often expressed as a percentage change per unit of time. The growth rate is a crucial concept in the study of exponential functions and the analysis of exponential and logarithmic equations.
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 Functions: An inverse function is a function that undoes the operation of another function. It reverses the relationship between the input and output values, allowing you to find the original input when given the output. Inverse functions are particularly important in the study of exponential and other types of equations.
Linear growth: Linear growth describes a situation where a quantity increases by a constant amount over equal intervals of time. It is represented mathematically by the equation $y = mx + b$, where $m$ is the rate of growth and $b$ is the starting value.
Logarithms: Logarithms are the inverse function of exponents, allowing us to express exponential relationships in a more linear form. They are a powerful mathematical tool that can be used to simplify complex calculations and analyze growth or decay patterns.
Mathematical Modeling: Mathematical modeling is the process of using mathematical concepts, principles, and techniques to represent and analyze real-world phenomena, systems, or problems. It involves creating a simplified, abstract representation of a complex situation to gain insights, make predictions, and inform decision-making.
Natural Exponential Function: The natural exponential function, denoted as $e^x$, is a fundamental function in mathematics that describes continuous exponential growth or decay. It is the base-$e$ exponential function, where $e$ is the mathematical constant approximately equal to 2.718. The natural exponential function is a crucial concept in the study of exponential functions and their graphs.
Newton's Law of Cooling: Newton's Law of Cooling is a fundamental principle that describes the rate of heat transfer between an object and its surrounding environment. It states that the rate of change of an object's temperature is proportional to the difference between the object's temperature and the temperature of its surroundings.
Nominal rate: The nominal rate is the interest rate stated on a loan or investment, not accounting for compounding within a specific period. It represents the percentage of interest charged or earned in one year without considering the effect of compounding.
Radioactive decay: Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This process is a key example of exponential decay, where the amount of radioactive substance decreases over time at a rate proportional to its current amount. Understanding radioactive decay is crucial for applications in fields like nuclear physics, radiometric dating, and medical imaging.
Transformations: Transformations refer to the processes of altering the position, size, shape, or orientation of a graph in a coordinate plane. They are crucial for understanding how different functions behave when subjected to changes such as translations, reflections, stretches, and compressions. By applying these transformations, one can gain insight into the properties of various types of functions and how they can be manipulated to produce new graphs.
Y-intercept: The y-intercept is the point at which a line or curve intersects the y-axis, representing the value of the dependent variable (y) when the independent variable (x) is zero. It is a crucial concept in understanding the behavior and properties of various mathematical functions and their graphical representations.
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Glossary
6.2 Graphs of Exponential Functions