📈College Algebra Unit 6 – Exponential and Logarithmic Functions

Key Concepts and Definitions

• Exponential functions defined by the equation $f(x) = a^x$, where $a$ is a positive constant not equal to 1 called the base and $x$ is the exponent or power
• Logarithmic functions are the inverse of exponential functions, denoted as $\log_a(x)$, where $a$ is the base, and $x$ is the argument
  • Common logarithms have a base of 10 and are written as $\log(x)$
  • Natural logarithms have a base of $e$ (approximately 2.718) and are written as $\ln(x)$
• The domain of an exponential function includes all real numbers, while the range is always positive
• The domain of a logarithmic function is all positive real numbers, and the range includes all real numbers
• The exponential constant $e$ is an irrational number (approximately 2.718) that is the base of the natural exponential function and natural logarithm
• Exponential growth occurs when a quantity increases by a constant factor over equal intervals of time (population growth)
• Exponential decay happens when a quantity decreases by a constant factor over equal intervals of time (radioactive decay)

Properties of Exponential Functions

• The product rule states that $a^m \cdot a^n = a^{m+n}$, simplifying the multiplication of exponential expressions with the same base
• The quotient rule for exponential functions is $\frac{a^m}{a^n} = a^{m-n}$, which simplifies the division of exponential expressions with the same base
• The power rule for exponential functions states that $(a^m)^n = a^{mn}$, allowing for the simplification of exponents raised to a power
• Exponential functions are always positive for positive bases, as they are defined for all real numbers and never produce a zero or negative value
• The exponential function $f(x) = a^x$ is increasing if $a > 1$ and decreasing if $0 < a < 1$
  • For example, the function $f(x) = 2^x$ is increasing, while $g(x) = (\frac{1}{2})^x$ is decreasing
• The y-intercept of an exponential function is always $(0, 1)$, as $a^0 = 1$ for any base $a$
• Exponential functions have a horizontal asymptote at $y = 0$, meaning the graph approaches but never reaches the x-axis as $x$ approaches negative infinity

Graphing Exponential Functions

• To graph an exponential function, start by plotting the y-intercept at $(0, 1)$
• Identify the base $a$ and determine if the function is increasing ($a > 1$) or decreasing ($0 < a < 1$)
• Plot additional points by choosing x-values and calculating the corresponding y-values using the exponential function $f(x) = a^x$
  • For example, to graph $f(x) = 2^x$, plot points such as $(1, 2)$, $(2, 4)$, $(-1, \frac{1}{2})$, and $(-2, \frac{1}{4})$
• Connect the plotted points with a smooth curve, keeping in mind the shape of the exponential function (increasing or decreasing)
• Sketch the horizontal asymptote at $y = 0$ to show the long-term behavior of the function
• Transformations of exponential functions include vertical shifts, horizontal shifts, reflections, and vertical stretches or compressions
  • A vertical shift is represented by $f(x) = a^x + k$, moving the graph up or down by $k$ units
  • A horizontal shift is represented by $f(x) = a^{x-h}$, moving the graph left or right by $h$ units

Introduction to Logarithms

• Logarithms are the inverse of exponential functions, allowing us to solve for the exponent in an exponential equation
• The logarithmic equation $\log_a(x) = y$ is equivalent to the exponential equation $a^y = x$, where $a$ is the base, $x$ is the argument, and $y$ is the exponent
• Common logarithms, denoted as $\log(x)$, have a base of 10 and are used in many scientific and engineering applications (Richter scale, decibels)
• Natural logarithms, denoted as $\ln(x)$, have a base of $e$ (approximately 2.718) and are used in mathematical modeling and calculus
• The change of base formula allows for converting between logarithms with different bases: $\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$, where $a$ and $b$ are positive bases not equal to 1
• Logarithms are undefined for non-positive arguments, as they are the inverse of exponential functions, which always produce positive values

Properties of Logarithmic Functions

• The product rule for logarithms states that $\log_a(MN) = \log_a(M) + \log_a(N)$, allowing for the simplification of the logarithm of a product
• The quotient rule for logarithms is $\log_a(\frac{M}{N}) = \log_a(M) - \log_a(N)$, which simplifies the logarithm of a quotient
• The power rule for logarithms states that $\log_a(M^r) = r \cdot \log_a(M)$, enabling the simplification of the logarithm of a power
• The zero exponent rule for logarithms is $\log_a(1) = 0$, as $a^0 = 1$ for any base $a$
• The logarithm of the base is always equal to 1: $\log_a(a) = 1$, as $a^1 = a$ for any base $a$
• Logarithmic functions have a vertical asymptote at $x = 0$, meaning the graph approaches positive or negative infinity as $x$ approaches 0 from the right
• The x-intercept of a logarithmic function is always $(1, 0)$, as $\log_a(1) = 0$ for any base $a$

Solving Exponential and Logarithmic Equations

• To solve exponential equations, isolate the exponential expression on one side of the equation and take the logarithm of both sides
  • For example, to solve $2^x = 8$, take the logarithm (base 2) of both sides: $\log_2(2^x) = \log_2(8)$, which simplifies to $x = 3$
• When solving logarithmic equations, isolate the logarithmic expression on one side and rewrite it as an equivalent exponential equation
  • For example, to solve $\log_3(x) = 4$, rewrite it as an exponential equation: $3^4 = x$, which simplifies to $x = 81$
• If the bases of the exponential or logarithmic expressions are different, use the change of base formula to convert them to a common base
• Be aware of the domain restrictions when solving exponential and logarithmic equations, as logarithms are undefined for non-positive arguments, and exponential functions always produce positive values
• When solving equations involving natural logarithms or common logarithms, use the properties of logarithms to simplify the expressions before solving
• Check your solutions by substituting them back into the original equation to ensure they satisfy the equality

Real-World Applications

• Exponential functions model population growth, as the growth rate is proportional to the current population size (bacteria, compound interest)
• Radioactive decay is modeled by exponential functions, as the decay rate is proportional to the amount of remaining radioactive material (carbon dating, half-life)
• Logarithmic scales are used to represent large ranges of values in a compact form (Richter scale for earthquake magnitudes, decibels for sound intensity, pH scale for acidity)
• Logarithms are used in computer science and information theory to quantify the amount of information and complexity of algorithms (binary search, sorting)
• Exponential and logarithmic functions are used in finance to calculate compound interest, present value, and future value of investments
• In physics, exponential functions describe the behavior of electrical circuits (capacitor discharge) and cooling or heating processes (Newton's law of cooling)
• Logarithmic spirals are found in nature, such as the shape of nautilus shells and the arrangement of seeds in sunflowers, demonstrating the presence of these functions in the natural world

Common Mistakes and Tips

• Remember that the base of an exponential or logarithmic function must be positive and not equal to 1
• Be careful when using the properties of logarithms, as the bases must be the same for the properties to apply
• When solving equations, check for extraneous solutions that may arise from manipulating the equation, and verify the solutions by substituting them back into the original equation
• Pay attention to the domain restrictions of logarithmic functions, as they are undefined for non-positive arguments
• When graphing exponential or logarithmic functions, consider the key features such as the y-intercept, x-intercept, asymptotes, and the behavior of the function as $x$ approaches positive or negative infinity
• Use the change of base formula when necessary to convert between logarithms with different bases
• Remember that the natural exponential function and natural logarithm are inverses of each other, as are the exponential function and logarithm with the same base
• Practice identifying and applying the properties of exponential and logarithmic functions in various contexts to reinforce your understanding of these concepts