📈AP Pre-Calculus Unit 2 – Exponential and Logarithmic Functions

Key Concepts and Definitions

• Exponential functions involve a constant base raised to a variable power and take the form $f(x) = b^x$, where $b$ is a positive real number not equal to 1
• Logarithmic functions are the inverse of exponential functions and take the form $f(x) = \log_b(x)$, where $b$ is the base and $x$ is a positive real number
• The natural base $e$ is an irrational constant approximately equal to 2.71828 and is used in natural exponential functions $f(x) = e^x$ and natural logarithmic functions $f(x) = \ln(x)$
• Exponential growth occurs when a quantity increases by a constant percent over equal intervals, resulting in an exponential function with a base greater than 1
• Exponential decay happens when a quantity decreases by a constant percent over equal intervals, resulting in an exponential function with a base between 0 and 1
• The half-life of an exponentially decaying quantity is the time it takes for the quantity to be reduced by half
• Doubling time is the time it takes for an exponentially growing quantity to double in value

Properties of Exponential Functions

• Exponential functions are always positive for positive bases and never equal zero
• The y-intercept of an exponential function is always $(0, 1)$
• For exponential functions with a base greater than 1, the function increases as x increases, demonstrating exponential growth
  • As x approaches positive infinity, the function values approach positive infinity
  • As x approaches negative infinity, the function values approach 0
• For exponential functions with a base between 0 and 1, the function decreases as x increases, demonstrating exponential decay
  • As x approaches positive infinity, the function values approach 0
  • As x approaches negative infinity, the function values approach positive infinity
• The domain of an exponential function is all real numbers, while the range is all positive real numbers
• Exponential functions have a horizontal asymptote at y = 0

Graphing Exponential Functions

• To graph an exponential function, start by plotting the y-intercept at $(0, 1)$
• Identify the base of the function to determine if the function represents growth (base > 1) or decay (0 < base < 1)
• Plot additional points by choosing x-values and calculating the corresponding y-values using the exponential function
• Connect the points with a smooth curve, keeping in mind the general shape of exponential growth or decay
• Sketch the horizontal asymptote at y = 0
• Transformations of exponential functions include:
  • Vertical shifts: $f(x) = b^x + k$ shifts the graph up by $k$ units if $k > 0$ or down by $|k|$ units if $k < 0$
  • Horizontal shifts: $f(x) = b^{x-h}$ shifts the graph right by $h$ units if $h > 0$ or left by $|h|$ units if $h < 0$
  • Reflections: $f(x) = -b^x$ reflects the graph across the x-axis

Introduction to Logarithms

• Logarithms are the inverse of exponential functions, meaning they "undo" exponentiation
• The logarithm of a number $x$ with base $b$ is the exponent to which $b$ must be raised to get $x$, written as $\log_b(x) = y$ if and only if $b^y = x$
• Common logarithms have a base of 10 and are written as $\log(x)$ without a subscript
• Natural logarithms have a base of $e$ and are written as $\ln(x)$
• The domain of a logarithmic function is all positive real numbers, while the range is all real numbers
• Logarithmic functions have a vertical asymptote at x = 0

Properties of Logarithmic Functions

• Logarithms have several important properties that allow for simplifying and solving equations:
  • Product property: $\log_b(MN) = \log_b(M) + \log_b(N)$
  • Quotient property: $\log_b(M/N) = \log_b(M) - \log_b(N)$
  • Power property: $\log_b(M^r) = r \cdot \log_b(M)$
• Change of base formula: $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$, where $a$ is any positive base
• The logarithm of 1 in any base is always 0: $\log_b(1) = 0$
• The logarithm of the base in any base is always 1: $\log_b(b) = 1$
• Logarithms can be used to solve exponential equations by applying the logarithm to both sides of the equation, using the fact that logarithms and exponents "cancel" each other out

Graphing Logarithmic Functions

• To graph a logarithmic function, start by drawing the vertical asymptote at x = 0
• Plot the x-intercept at $(1, 0)$, which is true for all logarithmic functions
• Choose positive x-values and calculate the corresponding y-values using the logarithmic function
• Plot the points and connect them with a smooth curve, keeping in mind the general shape of the logarithmic function
• Transformations of logarithmic functions include:
  • Vertical shifts: $f(x) = \log_b(x) + k$ shifts the graph up by $k$ units if $k > 0$ or down by $|k|$ units if $k < 0$
  • Horizontal shifts: $f(x) = \log_b(x - h)$ shifts the graph right by $h$ units if $h > 0$ or left by $|h|$ units if $h < 0$
  • Reflections: $f(x) = -\log_b(x)$ reflects the graph across the x-axis
• The graph of a logarithmic function is a reflection of the corresponding exponential function across the line y = x

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 $x = 3$
• To solve logarithmic equations, isolate the logarithmic expression on one side of the equation and rewrite it as an exponential equation
  • For example, to solve $\log_3(x) = 4$, rewrite it as an exponential equation: $3^4 = x$, which simplifies to $x = 81$
• When solving equations involving both exponential and logarithmic expressions, apply the properties of logarithms to simplify the equation before solving
• Be aware of the domain restrictions when solving equations, as logarithms are only defined for positive arguments

Real-World Applications

• Exponential functions can model population growth, compound interest, and radioactive decay
  • Population growth: $P(t) = P_0e^{rt}$, where $P_0$ is the initial population, $r$ is the growth rate, and $t$ is time
  • Compound interest: $A(t) = P(1 + r)^t$, where $P$ is the principal, $r$ is the interest rate per compounding period, and $t$ is the number of compounding periods
  • Radioactive decay: $A(t) = A_0e^{-kt}$, where $A_0$ is the initial amount, $k$ is the decay constant, and $t$ is time
• Logarithmic functions can model the Richter scale for earthquake magnitudes, the pH scale for acidity, and the decibel scale for sound intensity
  • Richter scale: $M = \log(\frac{I}{I_0})$, where $M$ is the magnitude, $I$ is the intensity of the earthquake, and $I_0$ is a reference intensity
  • pH scale: $pH = -\log[H^+]$, where $[H^+]$ is the concentration of hydrogen ions in a solution
  • Decibel scale: $\beta = 10\log(\frac{I}{I_0})$, where $\beta$ is the sound intensity level in decibels, $I$ is the sound intensity, and $I_0$ is a reference intensity
• Exponential and logarithmic functions are used in fields such as biology, chemistry, physics, economics, and computer science to model various phenomena and solve problems