Transcript Notes: Exponential vs. Logarithmic Functions and Notation

Key question from transcript: exponential functions have “unlimited domain” or “all real numbers”?
- Exponential functions have domain for all real numbers: ext{Domain} = (-
  \infty, \infty) = \mathbb{R}
- Example form: $y = a^x$ with base a>0, a\neq 1
Consequences and typical graph behavior
- For base a>1, the function is increasing; for 0<a<1, it is decreasing
- Range is always positive: $ext{Range} = (0, \infty)$ when a>0, a\neq 1
- Horizontal asymptote (not a vertical one): as $x\to -\infty$ , $y = a^x \to 0^+$ , so the horizontal asymptote is $y=0$
- Note: there is no vertical asymptote for exponential functions
Notation and real-number sets
- Real numbers are commonly denoted as $\mathbb{R}$ (blackboard bold R)
- In plain text, you might see simply R; in notes, we can use $\mathbb{R}$ or mention the symbol
Relationship to limits (context from transcript)
- In this graph-focused context, the instructor indicated no limits will be required for this particular graph
- Limits are a calculus concept; precalculus graphs here do not require limit calculations
Quick example to anchor concepts
- If $a=2$ , then $y=2^x$ has domain $(-\infty, \infty)$ , range $(0, \infty)$ , and horizontal asymptote at $y=0$ as $x\to -\infty$
- If $a=\tfrac{1}{2}$ , then $y=(1/2)^x$ has domain $(-\infty, \infty)$ , range $(0, \infty)$ , decreasing, with the same horizontal asymptote $y=0$ as $x\to -\infty$

Clarification from transcript
- The question about vertical asymptotes for exponentials is answered: exponentials do not have vertical asymptotes
- The vertical asymptote discussion is relevant to logarithmic functions
Key properties of logarithms (contextual, not directly from transcript but related to the question)
- Standard form: $y = \log_a x$ with base a>0, a\neq 1
- Domain: x>0; Range: $(-\infty, \infty)$
- Vertical asymptote at $x=0$ ; as $x\to 0^+$ , $\log_a x \to -\infty$
- As $x\to \infty$ , $\log_a x \to \infty$
- Inverse relationship: $\log_a x$ is the inverse of $a^x$
Summary relation between exponentials and logs
- Exponential: domain $\mathbb{R}$ , range $(0, \infty)$ , horizontal asymptote $y=0$ (as $x\to -\infty$ )
- Logarithm: domain $(0, \infty)$ , range $\mathbb{R}$ , vertical asymptote $x=0$
- They are inverse functions of each other: $\log<em>a (a^x) = x$ and $a^{\log</em>a x} = x$ for appropriate domains

The real number set is commonly denoted by $\mathbb{R}$ (double-struck R)
In plain text or less formal contexts, you may see simply R, but $\mathbb{R}$ is preferred in formal notes
In the context of the transcript, students may have asked about the symbol with a bar on the side; the standard is the double-struck $\mathbb{R}$

The student asked if limits will be required for the graphs
- Instructor’s answer: No, this graph section does not require taking limits
- Limits are a calculus topic; this content aligns more with precalculus concepts
Practical implications
- When sketching graphs of exponential and logarithmic functions in this context, focus on domain, range, growth/decay behavior, and asymptotes
- Do not expect limit computations for these particular graphs unless explicitly introduced in a later calculus unit

Exponential growth and decay appear in population models, compound interest, and biological processes; the domain being all real numbers reflects the ability to input any real time value
Logarithms are used in pH scales, decibel scales, Richter scales, and when dealing with multiplicative processes; the vertical asymptote at x=0 corresponds to the idea that very small inputs (approaching zero from the right) produce extremely large outputs in the log scale
Understanding the inverse relationship helps in solving equations where an exponent is inside a logarithm or vice versa

Exponential function basics
- $y = a^x$ with a>0, a\neq 1
- Domain: $(-\infty, \infty)$ ; Range: $(0, \infty)$
- Horizontal asymptote: $y=0$ as $x\to -\infty$
- Inverse relationship: $\log_a x$ is the inverse of $a^x$
Logarithmic function basics
- $y = \log_a x$ with a>0, a\neq 1
- Domain: x>0; Range: $(-\infty, \infty)$
- Vertical asymptote: $x=0$ ; as $x\to 0^+$ , $y\to -\infty$
- Inverse relationship: $\log_a x$ is the inverse of $a^x$
Real numbers notation
- $\mathbb{R} = {\text{all real numbers}}$

Note: If you want, I can convert this into a more condensed study sheet or expand any section with additional examples and practice problems.