8AD - 9AB - Exponential Functions Review & Logarithmic Functions Video lessons (Notes)

Review of Exponential Functions in Unit 89

Comparing Exponential Functions

In this lesson, we focus specifically on exponential functions, comparing two distinct types that demonstrate different behaviors based on their bases. One function is characterized by a base greater than 1, exemplified by (y = 2^x), while the other features a fractional base between 0 and 1, represented by (y = (1/2)^x). This comparative study of their graphs and behaviors serves as a recap of key concepts encountered in Grade 11 mathematics, enhancing understanding of function behavior in both growth and decay scenarios.

Graphing (y = 2^x)

To thoroughly investigate the nature of (y = 2^x), we generate a table of values spanning both negative and positive exponents. For example:

  • (2^{-3} = \frac{1}{2^3} = \frac{1}{8})

  • Continuing this process gives the following values:

    • (x = -3) results in (y = \frac{1}{8})

    • (x = -2) gives (y = \frac{1}{4})

    • (x = -1) yields (y = \frac{1}{2})

    • (x = 0) results in (y = 1)

    • (x = 1) returns (y = 2)

    • (x = 2) brings us to (y = 4)

    • (x = 3) to (y = 8)

    • and finally (x = 4) results in (y = 16). After obtaining these values, we plot them to graph the exponential function. The resulting graph reveals an increasing trend, which approaches the x-axis asymptotically without crossing it, indicating that the curve nears (y = 0). This phenomenon represents a horizontal asymptote at (y = 0), showing that as (x) approaches negative infinity, (y) approaches zero, highlighting the function's growth over positive x-values.

Graphing (y = (1/2)^x)

Next, we analyze the function (y = (1/2)^x). The preparatory work for plotting this graph yields:

  • ((1/2)^{-3} = 2^3 = 8)

  • Similar calculations provide the following values:

    • (x = -2) results in (y = 4)

    • (x = -1) yields (y = 2)

    • (x = 0) gives (y = 1)

    • (x = 1) returns (y = \frac{1}{2})

    • (x = 2) yields (y = \frac{1}{4})

    • (x = 3) gives (y = \frac{1}{8})

    • and finally (x = 4) results in (y = \frac{1}{16}). When these points are plotted, an exponential decay curve emerges, where the graph again approaches 0 without crossing the x-axis, confirming the presence of a horizontal asymptote at (y = 0). This function's behavior illustrates how exponential decay contrasts sharply with growth.

Key Observations on Both Functions

  • X-Intercepts: Both functions do not intercept the x-axis; they asymptotically approach the line (y = 0) without touching it.

  • Y-Intercepts: Each function crosses the y-axis at (y = 1), occurring at the point where (x = 0).

  • Domain: Both functions have a domain of all real numbers, denoted as (x \in R).

  • Range: The range for each function consists of positive values, indicated as (y > 0).

  • End Behavior: For (y = 2^x), as (x) approaches positive infinity, (y) approaches positive infinity; while as (x) approaches negative infinity, (y) approaches 0. Conversely, for (y = (1/2)^x), as (x) approaches positive infinity, (y) approaches 0, while as (x) approaches negative infinity, (y) approaches positive infinity.

Characteristics of Exponential Functions

These characteristics define two types of exponential functions:

  • (y = b^x) where (b > 1) yields an increasing function, a behavior exemplified by the graph of (y = 2^x).

  • (y = b^x) where (0 < b < 1) results in a decreasing function, as shown in (y = (1/2)^x). All conclusions about intercepts, asymptotes, and behaviors apply to functions fitting these definitions, foundational in the study of exponential growth and decay.

Introduction to Logarithmic Functions

Transitioning from exponential to logarithmic functions, we recognize the logarithm as the inverse of an exponential function. The logarithmic function can be expressed in the form: [ y = \log_b(x) ] where (b) is the base of the logarithm. In exponential terms, if (y = b^x), then rearranging this leads to (x = b^y), which translates to logarithmic form as (y = \log_b(x)).

Evaluating Logs without a Calculator

Example 1: Evaluating (\log_{36}(36))

To evaluate this logarithm, we express it in exponential form: Set it as (x = \log_{36}(36)), which translates to the equation (36^x = 36). Thus, it implies (x = 1) since the exponent that yields 36 when raising 36 is simply 1.

Example 2: Evaluating (\log_{(1/4)}(8))

Utilizing similar techniques, set it as (x = \log_{(1/4)}(8)), which leads to ((1/4)^x = 8). Recognizing that we can express both in terms of base 2 gives us ((2^{-2})^x = 2^3), which simplifies to (-2x = 3) and thus (x = -\frac{3}{2}). This example demonstrates the ability to transition between formats effectively.

Transitioning between Log Form and Exponential Form

When converting log form to exponential form, it is essential to maintain consistency in the bases and exponents for accuracy and clarity.

Graphing Logarithmic Functions

To graph the function (y = \log_2(x)), compute values for its inverse (y = 2^x). This inversion allows for a simplified approach to data generation: Using points like ((-1, \frac{1}{2})), ((0, 1)), ((1, 2)), and ((2, 4)) helps to define the growth curve of the exponential function, which aids in plotting logarithmic functions effectively.

Asymptotes and Intercepts

The domain of logarithmic functions includes (x > 0) with a range of all real numbers. It features a vertical asymptote at (x = 0) (y-axis) with no y-intercept, achieving a value of 0 at the point ((1, 0)).

Laws of Logarithms

Three essential laws of logarithms to employ are:

  • Logarithm of a Product: (\log_b(m*n) = \log_b(m) + \log_b(n))

  • Logarithm of a Quotient: (\log_b(m/n) = \log_b(m) - \log_b(n))

  • Logarithm of a Power: (\log_b(m^n) = n*\log_b(m))

Practical Application of the Log Laws

When evaluating logarithmic expressions, applying these laws can greatly simplify the process of logging or evaluating complex expressions.

Conclusion

In summary, today's lesson revisits the profound properties of both exponential and logarithmic functions, reinforcing key concepts, methods of evaluation, graphing methodologies, and the practical applications of logarithmic laws. This knowledge sets a solid foundation for deeper exploration in upcoming units B-D, essential for advanced mathematics pursuits.