Exponential and Logarithmic Fns

Exponential and Logarithmic Functions

4.1 Exponential Functions

• Definition: A function is of the form f(x) = a^x, where a > 0 and a ≠ 1. The base 'a' determines the function's characteristics.

• Domain: The domain is all real numbers: Df = ℝ.

• Behavior Based on Base 'a':

  • If a > 1, the function is increasing, demonstrating exponential growth.

  • If 0 < a < 1, the function is decreasing, indicating exponential decay.

• Key Points on the Graph:

  • (-1, 1/a): Point where the function values are decreasing (for a > 1) or increasing (for a < 1).

  • (0, 1): This point indicates the y-intercept, which is consistent across all exponential functions.

  • (1, a): This point represents how the function value changes based on the base 'a'.

Properties of Exponential Functions

• Domain: Df = ℝ

• Range: Rf = (0, +∞) indicating that all outputs are positive.

• Intercepts:

  • No x-intercepts: Exponential functions never touch the x-axis, confirming that they are always positive.

  • y-intercept: located at (0, 1), showing the function's value when x = 0.

• Horizontal Asymptote: The graph approaches the x-axis (y = 0) but never intersects it.

• Increasing/Decreasing Behavior: The function is increasing where a > 1 and decreasing where 0 < a < 1.

• One-to-One Function: This characteristic signifies that each output corresponds to exactly one input, which allows for an inverse function: g(x) = log_a(x).

• Equivalence Property: If a^u = a^v, then u = v, allowing for solving equations of this type.

Natural Exponential Function

• Defined as f(x) = e^x, where 'e' is an irrational number approximately equal to 2.71828. This function exhibits unique properties consistent with the characteristics of exponential growth and decay, such as continuous growth without bounds.

Transformations of Exponential Functions

• Example 1: Graphing f(x) = 3 - x - 2

  • Starting with the basic function y = 3^x, the graph is reflected about the y-axis to obtain y = 3^{-x}, then a vertical shift downward by 2 units is applied.

• Example 2: Graphing f(x) = 3e^{2x-1}

  • Begin with the basic function y = e^x, shift the graph to the right by 1 unit to yield y = e^{x-1}, apply horizontal compression, and vertical stretch based on the coefficient of x.

4.2 Logarithmic Functions

• Definition: A logarithmic function is of the form f(x) = log_a(x), where a > 0, a ≠ 1, and x > 0. It represents the inverse of exponential functions.

• Inverse Relationship: The relationship is defined as y = log_a(x) if and only if a^y = x. This highlights how logarithmic and exponential functions are interconnected.

Properties of Logarithmic Functions

I. Logarithm Definition: log_a(x) gives the exponent to which base 'a' must be raised to retrieve x.

• Example: For log2(8), this translates to determining 2^x = 8, concluding that log2(8) = 3 since 2 raised to the power of 3 equals 8. II. Change to Exponential Form:

• Example: log2(x) = 4 translates to 2^4 = x leading to x = 16, demonstrating the conversion process from logarithmic to exponential form. III. Domain & Range:

• Domain = (0, ∞), indicating that logarithmic functions can only accept positive x values.

• Range = (-∞, ∞), suggesting that output can span all real numbers. IV. Graph and Symmetry: The graph of f(x) = log_a(x) reflects symmetry with the graph of y = a^x across the line y = x, illustrating their inverse nature. V. Important Points on Graph: Includes key points such as (1/a, -1), (1, 0), and (a, 1), which are critical for sketching logarithmic functions. VI. X-Intercept: Found at (1, 0), confirming where the logarithmic function crosses the x-axis. VII. Vertical Asymptote: Occurs at the y-axis (x = 0), indicating the function's behavior as it approaches this line but never touching it. VIII. Function Behavior:

• Increasing for a > 1 and decreasing for 0 < a < 1, reflecting similar characteristics to the base exponential behavior. IX. One-to-One Property: If log_a(u) = log_a(v), then it is confirmed that u = v, reinforcing the function's unique mapping properties.

Additional Examples and Applications

• When defining the domain where logarithms are valid, it is crucial that the argument inside the logarithm is positive. For instance, solving equations like log5(x^2 + x + 4) = 2 involves determining valid input values through the polynomial's roots, converting to exponential form, and finally identifying solutions through interval analysis.

Summarizing Properties of Logarithms

• Key properties to remember include:

  • Basic Log Values: log_a(1) = 0 and log_a(a) = 1 to establish foundational logarithmic values.

  • Exponential Value of Natural Log: Effective application shows e^ln(4) = 4.

  • Log of Powers: Expressively simplified as log_a(M^r) = r * log_a(M).

  • Product Rule: log_a(M * N) = log_a(M) + log_a(N), allowing for combination of terms.

  • Use of Base Change: Conversions like log_a(M/N) = log_a(M) - log_a(N) assist in rearranging and simplifying logarithmic expressions.

Further Practice

• Students are encouraged to tackle systems involving logarithmic properties and exponential manipulation to find solutions. Understanding domain definitions for logarithmic functions as well as solving for variables through conversion between logarithmic and exponential forms play a crucial role in mastering these concepts.