Chapter 4 - Exponential and Logarithmic Functions

4.1 Exponential Functions

Definition of the Exponential Function

• The exponential function, denoted as $f(x) = b^x$, where:

  • $b$ is a positive constant (i.e., $b > 0$ and $b \neq 1$).

  • $x$ is any real number.

• In practical applications:

  • $a$ = initial amount (e.g., starting population, investment amount).

  • $b$ = growth (if $b > 1$) or decay factor (if $0 < b < 1$).

  • $x$ = time intervals (e.g., years, days, months).

• Key Applications: Often used to model scenarios like:

  • Bacterial growth/decay

  • Population dynamics

  • Financial applications such as compound interest.

Example: Evaluating an Exponential Function

• The function models average spending data: $f(x) = 42.2(1.56)^x$

• To evaluate for $x = 3$:

  • Substitute into the function:

  • $f(3) = 42.2(1.56)^3$

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

Graphing an Exponential Function

• Graphing Approach: Similar to polynomial functions, but consider the specific behavior of exponential functions.

  • The graph indicates that points are close to a horizontal asymptote or rapidly increase.

  • Key characteristics include understanding growth vs. decay:

  • Exponential Growth:

    • Form: $y = ab^x$ where $a > 0$ and $b > 1$.

    • Example: $f(x) = 2^x$ is upward curving.

  • Exponential Decay:

    • Form: $y = ab^x$ with $a > 0$ and $0 < b < 1$.

    • Example: $f(x) = (1/2)^x$ is downward curving.

Specific Points on Graphs

• When graphing, identify reasonable points.

• Fill in the graph's gaps based on general behavior and trend:

  • Example for Growth Graphing (e.g., $x = -3$ to $3$).

  • Example for Decay Graphing (e.g., $x = -3$ to $3$).

4.2 Logarithmic Functions

Objectives

• Change between logarithmic and exponential forms.

• Evaluate logarithmic expressions.

• Graph logarithmic functions.

• Identify the domain of logarithmic functions.

• Utilize common and natural logarithms.

Definition of the Logarithmic Function

• For $x > 0$ and $b > 0$, $b \neq 1$, the logarithmic function is defined as:

  • $y = \log_b x$ if and only if $b^y = x$.

Important Properties

1. One-to-One Functions: Only one-to-one functions can be inverted.

   • A function $f$ has an inverse if no horizontal line intersects its graph more than once.

2. Inverse Relationship: If $f(a) = b$, then $f^{-1}(b) = a$. The domain of $f$ corresponds to the range of its inverse.

3. Graphical Reflection: The graph of the inverse function is a reflection across the line $y = x$.

Evaluating Logarithmic Functions

• Convert from logarithmic to exponential form:

  • Example: $\log_4 26 = y$ leads to $4^y = 26$.

• Conversion affects the way we interpret and compute logarithmic values.

Domain of Logarithmic Functions

• The domain of a logarithmic function like $f(x) = \log x$ consists of all positive real numbers, i.e., $(0, ∞)$.

• Example: To find the domain of $f(x) = \log_4(x - 5)$, set up inequalities:

  • $x - 5 > 0$, which leads to $x > 5$.

Application of Logarithmic Functions

• Logarithmic properties include:

  • $\log_b 1 = 0$, since any base raised to the zero equals one.

  • $\log_b b = 1$ for any base $b$.

  • $\log_b b^x = x$ shows how logarithms can reverse exponentiation.

Common and Natural Logarithms

• Common logarithm (base 10) denoted as $\log x$.

• Natural logarithm (base e) denoted as $\ln x$.

4.3 Properties of Logarithms

Product Rule

• For positive real numbers $b, M, N$ (where $b \neq 1$), the logarithm of a product is:
  $\log<em>b (MN) = \log</em>b M + \log_b N$.

• Example: $\ln(7x) = \ln 7 + \ln x$.

Quotient Rule

• The logarithm of a quotient is:
  $\log<em>b (\frac{M}{N}) = \log</em>b M - \log_b N$.

Power Rule

• For any real number $p$, the logarithm of a number with an exponent is:
  $\log<em>b (M^p) = p \log</em>b M$.

Change of Base Formula

• For any logarithmic bases $a, b$ and any positive number $M$, it holds: $\log<em>b M = \frac{\log</em>a M}{\log_a b}$.

  • Use this to evaluate logarithms using scientific calculators or logarithmic tables.

Conclusion

• Familiarity with exponential and logarithmic functions, properties, and how to graph them is crucial in algebra and its applications in real-world contexts.