Notes on Exponential and Logarithmic Functions

Exponential Functions

• General Form: $f(x) = b^x$, where $b$ is the base.
• Parent Function Transformations: If we have the parent function, we can perform transformations such as reflections about the y-axis and vertical shifts.
  • Example: Reflecting about the y-axis and shifting 2 units upward.
• Domain: The domain of an exponential function is all real numbers, denoted as $R$.
• Range: The range is $(2, \infty)$.
• Property of Equality: If the bases are equal, then the exponents are equal, and vice versa.

Key Points of Exponential Functions

• Domain: The set of all real numbers, $R$.
• Range: The set of positive real numbers, $R^+$.
• x-intercept: Does not exist.
• y-intercept: The point $(0, 1)$.
• Key points: $(0, 1)$, $(1, b)$, and $(-1, \frac{1}{b})$.

Solving Exponential Equations

• Example 1:
  • $3^{6x-1} = 9^2$
  • $3^{6x-1} = (3^2)^2$
  • $3^{6x-1} = 3^4$
  • $6x - 1 = 4$
  • $x = \frac{5}{6}$
• Example 2:
  • $25^{1-x} = 5^{7-x}$
  • $(5^2)^{1-x} = 5^{7-x}$
  • $5^{2(1-x)} = 5^{7-x}$
  • $2 - 2x = 7 - x$
  • $x = -5$

Solving Exponential Inequalities

• When b > 1, the function is increasing.
• When 0 < b < 1, the function is decreasing.

Examples

• Example 1:
  • 2^{x+2} > \frac{1}{32}
  • 2^{x+2} > 2^{-5}
  • Since the base is greater than 1, the inequality sign remains the same: x + 2 > -5
  • x > -7
• Example 2:
  • 16^{2x-3} < 2^8
  • (2^4)^{2x-3} < 2^8
  • 2^{8x-12} < 2^8
  • Since the base is greater than 1: 8x - 12 < 8
  • 8x < 20
  • x < \frac{20}{8} = \frac{5}{2}

Exponential Growth and Decay Functions

• Exponential Growth: When b > 1, the function represents exponential growth.
• Exponential Decay: When 0 < b < 1, the function represents exponential decay.

Key Points

• (Growth): The point $(0, 1)$ and $(1, b)$.
• (Decay): The point $(0, 1)$ and $(1, b)$.
• Increasing/Decreasing: Exponential growth functions are increasing, while exponential decay functions are decreasing.

Logarithmic Functions

Definition of Logarithm

• The logarithm of $x$ to the base $b$ is the exponent to which $b$ must be raised to produce $x$.
• Logarithmic Form: $y = \log_b(x)$
• Exponential Form: $x = b^y$
• Where x > 0, and b > 0 and $b \neq 1$.

Key Concepts

• Converting Between Logarithmic and Exponential Forms
• Evaluating Logarithmic Expressions

Key Points of Logarithmic Functions

• Domain: The set of all real numbers greater than zero, i.e., $(0, \infty)$.
• Range: The set of all real numbers, $R$.
• x-intercept: The point $(1, 0)$.
• y-intercept: Does not exist.

General Form of Logarithmic Functions

• $f(x) = \log_b(x)$

Finding Domain and Asymptotes

• Domain: Find the values of $x$ for which the argument of the logarithm is greater than zero.
• Vertical Asymptote: Set the argument of the logarithm equal to zero and solve for $x$.
• y-intercept: Substitute $x = 0$ into the function.
• x-intercept: Set the function equal to zero and solve for $x$.

Example of Finding Domain and Vertical Asymptote

• Given the function: $f(x) = \log(x-a)$
• Domain: x - a > 0 \implies x > a, so the domain is $(a, \infty)$.
• Vertical Asymptote: $x - a = 0 \implies x = a$

Applications of Exponential Growth

• Exponential Growth Formula: $A = P(1 + r)^t$
  • Where:
    • $A$ = the amount or number after $t$ years.
    • $P$ = the initial amount or number.
    • $r$ = the annual growth rate (as a decimal).
    • $t$ = the number of years.

Example: Population Growth

• If a school has 2000 students initially and the student population increases by a fixed percentage annually, we can use the exponential growth formula to determine the number of students after a certain number of years.

Logarithmic functions vs Exponential function

• The logarithmic function is the inverse of the exponential function.
• Exponential growth functions increase, while exponential decay functions decrease.
• Range of exponential function: $(0, \infty)$.
• Domain of logarithmic function: $(0, \infty)$.