In-Depth Notes on Exponential and Logarithmic Functions

Exponential Functions

  • Definition: The exponential function with base $b$ is defined as:

    • f(x)=bxf(x) = b^x

    • Where $b$ is a positive constant ($b > 0$ and b≠1b \neq 1).


Evaluating Exponential Functions

  • Example:

    • Function: f(x)=42.2(1.56)xf(x) = 42.2(1.56)^x

    • Task: Find average spent after 3 hours, i.e., evaluate f(3)f(3):

    • f(3)=42.2(1.56)3f(3) = 42.2(1.56)^3

    • Calculation:

      • Approx. result: f(3)≈160f(3) \approx 160

    • Conclusion: Average amount spent after 3 hours is approximately $160.

Graphing Exponential Functions

  • Example: Graph f(x)=3xf(x) = 3^x:

    • Points to calculate:

    • f(−2)=3−2=19f(-2) = 3^{-2} = \frac{1}{9}

    • f(−1)=3−1=13f(-1) = 3^{-1} = \frac{1}{3}

    • f(0)=30=1f(0) = 3^{0} = 1

    • f(1)=31=3f(1) = 3^{1} = 3

    • Plot points: (-2, 1/9), (-1, 1/3), (0, 1), (1, 3)

    • Connect these points with a smooth curve.

Characteristics of Exponential Functions (Form: f(x)=bxf(x) = b^x)

  1. Domain and Range:

    • Domain: all real numbers, (−∞,∞)(-\infty, \infty)

    • Range: Positive real numbers, (0,∞)(0, \infty)

  2. Graph Interception:

    • Passes through (0,1); no x-intercept.

  3. Behavior Based on $b$:

    • If $b > 1$, the function increases, steeper with larger $b$.

    • If $0 < b < 1$: The function is decreasing, steeper with smaller $b$.

  4. One-to-One:

    • The function is one-to-one and has an inverse that is a function.

  5. Asymptotic Behavior:

    • Graph approaches the x-axis but does not touch it (horizontal asymptote at $y=0$).

Transformations of Exponential Functions

  • Vertical Transformations:

    • f(x)=g(x)+cf(x) = g(x) + c: Shift upward by $c$ units.

    • f(x)=g(x)−cf(x) = g(x) - c: Shift downward by $c$ units.

  • Horizontal Transformations:

    • f(x)=g(x+c)f(x) = g(x + c): Shift left by $c$ units.

    • f(x)=g(x−c)f(x) = g(x - c): Shift right by $c$ units.

  • Reflections:

    • f(x)=−g(x)f(x) = -g(x): Reflect about x-axis.

    • f(x)=g(−x)f(x) = g(-x): Reflect about y-axis.

  • Vertical Stretch/Shrink:

    • f(x)=cg(x)f(x) = c g(x): Stretch if $c > 1$, shrink if $0 < c < 1$.

  • Horizontal Stretch/Shrink:

    • f(x)=g(cx)f(x) = g(cx): Shrinks if $c > 1$, stretches if $0 < c < 1$.

The Natural Base $e$

  • Defined as the value that (1+1n)n\left(1 + \frac{1}{n}\right)^n approaches as $n$ increases.

  • Approximation: e≈2.718281827e \approx 2.718281827.

  • The function is called the natural exponential function f(x)=exf(x) = e^x.

Applications: Gray Wolf Population

  • Model: f(x)=1144⋅0.0325xf(x) = 1144 \cdot 0.0325^x for years after 1978.

  • Projection: To find the population in 2014 ($x = 36$):

    • Substitute and evaluate:

    • f(36)=1144⋅0.032536≈3686f(36) = 1144 \cdot 0.0325^{36} \approx 3686

    • Conclusion: Projected population of gray wolves in 2014 is about 3686.

Compound Interest Formulas

  • After $t$ years:

    • For $n$ compounding periods per year:

    • A=P(1+rn)ntA = P\left(1 + \frac{r}{n}\right)^{nt}

    • For continuous compounding:

    • A=PertA = Pe^{rt}.