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Calculus I
Table of Contents
Unit 1 Overview: Functions and Graphs
1.1 Review of Functions
1.2 Basic Classes of Functions
1.3 Trigonometric Functions
1.4 Inverse Functions
1.5 Exponential and Logarithmic Functions

∫calculus i review

1.5 Exponential and Logarithmic Functions

Citation:

Exponential and logarithmic functions are powerful tools for modeling growth, decay, and complex relationships. They're inverses of each other, with exponentials growing rapidly and logarithms slowing down as x increases.

These functions pop up everywhere, from compound interest to population dynamics. Understanding their properties and rules is key to solving real-world problems and grasping more advanced math concepts.

Exponential Functions

Graphing exponential functions

  • General form $f(x) = a \cdot b^x$ represents exponential functions
    • Vertical stretch factor and y-intercept determined by $a$ (0, a)
    • Base of the exponential function given by $b$
      • Function increases (grows) as x increases when $b > 1$ (2, e)
      • Function decreases (decays) as x increases when $0 < b < 1$ (1/2, 1/e)
  • Exponential growth modeled by $f(x) = a \cdot (1 + r)^x$ with growth rate $r > 0$ (5%, 0.1)
  • Exponential decay represented by $f(x) = a \cdot (1 - r)^x$ with decay rate $0 < r < 1$ (2%, 0.05)
    • Half-life is the time required for a quantity to reduce to half its initial value in exponential decay
  • Horizontal asymptote $y = 0$ exists for exponential functions with $0 < b < 1$

Comparison of exponential bases

  • Exponential functions with $b > 1$ always increase
    • Steeper growth results from larger bases (2 vs 3, e vs 10)
  • Exponential functions with $0 < b < 1$ always decrease
    • Steeper decay caused by smaller bases (1/2 vs 1/3, 1/e vs 0.1)
  • Point (0, 1) is common to all exponential functions

Significance of natural base e

  • Mathematical constant $e \approx 2.71828$ known as the natural base
  • Natural exponential functions $f(x) = e^x$ have base $e$
    • Model continuous growth or decay processes (population, radioactivity)
  • Applications of natural exponential functions include:
    • Continuous compound interest (bank accounts, investments)
    • Population growth models (bacteria, viral spread)
    • Radioactive decay (carbon dating, nuclear physics)

Logarithmic Functions

Logarithmic functions and graphs

  • General form $f(x) = \log_b(x)$ represents logarithmic functions, where $b > 0$ and $b \neq 1$
    • Logarithm $\log_b(x)$ gives the exponent to which $b$ must be raised to get $x$
    • Common logarithm (base 10) is written as $\log(x)$ or $\log_{10}(x)$
    • Natural logarithm (base e) is denoted as $\ln(x)$ or $\log_e(x)$
  • Domain restricted to $x > 0$ as logarithms only defined for positive real numbers
  • Range includes all real numbers
  • Vertical asymptote occurs at $x = 0$
  • Logarithmic functions serve as the inverse of exponential functions

Exponential vs logarithmic functions

  • Inverse relationship: If $y = b^x$, then $x = \log_b(y)$
    • Example: If $y = 2^x$, then $x = \log_2(y)$
  • Graphically, exponential and logarithmic functions reflect each other over the line $y = x$

Change of base in logarithms

  • Change of base formula $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$ allows calculation of logarithms with any base
    • Requires $a > 0$, $a \neq 1$, and $b > 0$, $b \neq 1$
    • Enables use of common bases (10, e) on calculators for logarithms with different bases

Hyperbolic Functions

Properties of hyperbolic functions

  • Hyperbolic sine $\sinh(x) = \frac{e^x - e^{-x}}{2}$
    • Odd function with domain and range of all real numbers
  • Hyperbolic cosine $\cosh(x) = \frac{e^x + e^{-x}}{2}$
    • Even function with domain of all real numbers and range $y \geq 1$
  • Hyperbolic tangent $\tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$
    • Odd function with domain of all real numbers and range $-1 < y < 1$
    • Horizontal asymptotes at $y = 1$ and $y = -1$
  • Hyperbolic functions analogous to trigonometric functions but defined using exponential functions instead of circular functions

Exponent and Logarithm Rules

Exponent Rules

  • Product rule: $a^m \cdot a^n = a^{m+n}$
  • Quotient rule: $\frac{a^m}{a^n} = a^{m-n}$
  • Power rule: $(a^m)^n = a^{mn}$
  • Zero exponent rule: $a^0 = 1$ (for $a \neq 0$)
  • Negative exponent rule: $a^{-n} = \frac{1}{a^n}$

Logarithm Rules

  • Product rule: $\log_b(xy) = \log_b(x) + \log_b(y)$
  • Quotient rule: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$
  • Power rule: $\log_b(x^n) = n\log_b(x)$
  • Change of base: $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$

Solving Exponential and Logarithmic Equations

  • Exponential equations often require logarithms to solve
  • Logarithmic equations may require exponentials to solve
  • Use inverse functions and properties to isolate variables

Key Terms to Review (24)

Base: The base in an exponential function is the constant value that is raised to a variable exponent. In logarithmic functions, the base is the constant value that the logarithm operates on.
Compounding interest: Compounding interest is the process where the value of an investment increases because the earnings on an investment, both capital gains and interest, earn interest as time passes. This exponential growth can be modeled using functions involving exponents.
Common logarithm: A common logarithm is a logarithm with base 10, often denoted as $\log_{10}$ or simply log. It is used to solve equations involving exponential growth or decay where the base of the exponent is 10.
Exponent: An exponent indicates how many times a number, known as the base, is multiplied by itself. It is typically represented as a small number to the upper right of the base.
Domain: The domain of a function is the set of all possible input values (typically $x$-values) for which the function is defined. It represents all the values that can be plugged into the function without causing any undefined behavior.
Earthquake: An earthquake is a sudden and violent shaking of the ground caused by movements within the Earth's crust. It results in seismic waves that propagate through the Earth.
Logarithmic function: A logarithmic function is the inverse of an exponential function and is typically written as $y = \log_b(x)$, where $b$ is the base. It represents the power to which the base must be raised to obtain a given number.
Hyperbolic functions: Hyperbolic functions are analogs of trigonometric functions but for a hyperbola rather than a circle. They include $\sinh(x)$, $\cosh(x)$, $\tanh(x)$, and their reciprocals.
Inverse function: An inverse function is a function that reverses the effect of the original function. If $f(x)$ is a function, then its inverse $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.
Horizontal asymptote: A horizontal asymptote of a function is a horizontal line that the graph of the function approaches as x tends to positive or negative infinity. It indicates the behavior of the function at extreme values of x.
Inverse hyperbolic functions: Inverse hyperbolic functions are the inverses of the hyperbolic functions, such as sinh, cosh, and tanh. They are used to solve equations involving hyperbolic functions.
Population growth: Population growth describes the change in the number of individuals in a population over time. It can be modeled using exponential and logarithmic functions to predict future changes.
Natural logarithm: The natural logarithm, denoted as $\ln(x)$, is the logarithm to the base $e$, where $e$ is an irrational constant approximately equal to 2.71828. It is the inverse function of the exponential function $e^x$.
Natural exponential function: The natural exponential function is defined as $e^x$, where $e$ is Euler's number, approximately equal to 2.71828. It is a fundamental function in calculus with unique properties related to growth and decay.
Richter scale: The Richter scale is a logarithmic scale used to quantify the magnitude of an earthquake. It measures the amount of energy released by an earthquake on a base-10 scale.
Vertical asymptote: A vertical asymptote is a line $x = a$ where the function $f(x)$ approaches positive or negative infinity as $x$ approaches $a$. Vertical asymptotes occur at values of $x$ that make the denominator of a rational function zero, provided that the numerator does not also become zero at those points.
Logarithmic Function: A logarithmic function is a mathematical function that describes an exponential relationship between two quantities. It is the inverse of an exponential function, allowing for the representation of quantities that grow or decay at a constant rate over time. Logarithmic functions are essential in various fields, including mathematics, science, and engineering, for their ability to model and analyze complex phenomena.
Exponential Function: An exponential function is a mathematical function where the variable appears as the exponent. These functions exhibit a characteristic pattern of growth or decay, making them important in various fields of study, including calculus, physics, and finance.
Y-intercept: The y-intercept is the point at which a graph or function intersects the y-axis, representing the value of the function when the input (x-value) is zero. It is a crucial concept in understanding the behavior and properties of various functions, including linear, exponential, and logarithmic functions.
Domain: The domain of a function refers to the set of all possible input values for which the function is defined. It represents the range of values that the independent variable can take on, and it is a crucial concept in understanding the behavior and properties of functions.
Horizontal Asymptote: A horizontal asymptote is a horizontal line that a function's graph approaches as the input variable (usually x) approaches positive or negative infinity. It represents the limiting value that the function approaches but never quite reaches.
Vertical Asymptote: A vertical asymptote is a vertical line that a function's graph approaches but never touches. It represents the value of the independent variable where the function becomes undefined or experiences a vertical discontinuity.
Inverse Function: An inverse function is a function that reverses the relationship between the input and output of another function. It takes the output of the original function and produces the corresponding input, effectively undoing the original function's operation.
Range: Range refers to the set of all possible output values (or 'y' values) that a function can produce based on its domain (the set of input values). Understanding the range helps us grasp how a function behaves, what outputs are attainable, and the limitations on those outputs.