Precalculus Exponential and Logarithmic Functions Syllabus

Exponential and Logarithmic Functions syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Exponential and Logarithmic Functions in Precalculus, not a summary of it.

1. 1

   Exponential Functions

   3 topics

   • Graphs of Exponential Functions
   • The Natural Base e
   • Transformations of Exponential Graphs
2. 2

   Logarithmic Functions

   3 topics

   • Logarithm as an Inverse
   • Common and Natural Logarithms
   • Graphs of Logarithmic Functions
3. 3

   Properties of Logarithms

   3 topics

   • Product, Quotient, and Power Rules
   • Change-of-Base Formula
   • Expanding and Condensing Logarithms
4. 4

   Exponential and Logarithmic Equations

   3 topics

   • Solving Exponential Equations
   • Solving Logarithmic Equations
   • Extraneous Solutions
5. 5

   Modeling and Applications

   4 topics

   • Compound Interest
   • Exponential Growth and Decay
   • Logistic Growth Models
   • Newton's Law of Cooling

Exponential and Logarithmic Functions flashcards for Precalculus

24 of 50 cards from the Exponential and Logarithmic Functions deck — real questions with worked answers.

1. What is the general form of an exponential function, and what restrictions apply to the base $b$?

   $f(x) = b^{x}$, where $b > 0$ and $b \neq 1$. (More generally $f(x) = a\,b^{x}$ with $a \neq 0$.)

2. For the exponential function $f(x) = b^{x}$, state its domain and range.

   Domain: $(-\infty, \infty)$. Range: $(0, \infty)$.

3. How does the graph of $f(x) = b^{x}$ behave when $b > 1$ versus $0 < b < 1$?

   If $b > 1$ the graph shows exponential growth (increasing). If $0 < b < 1$ it shows exponential decay (decreasing).

4. What is the horizontal asymptote of the basic exponential graph $f(x) = b^{x}$, and what point does every such graph pass through?

   Horizontal asymptote $y = 0$ (the $x$-axis); every graph passes through $(0, 1)$ since $b^{0} = 1$.

5. Approximately what is the value of the natural base $e$, and how is it defined as a limit?

   $e \approx 2.71828$, defined as $e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n}$.

6. What is the natural exponential function, and why is $e$ especially important in calculus?

   $f(x) = e^{x}$. It is its own derivative: $\frac{d}{dx}e^{x} = e^{x}$, making it the natural choice for growth/decay modeling.

7. Evaluate $e^{0}$ and describe the $y$-intercept and asymptote of $f(x) = e^{x}$.

   $e^{0} = 1$, so the $y$-intercept is $(0, 1)$; the horizontal asymptote is $y = 0$.

8. Given $f(x) = b^{x}$, what transformation does $f(x) + k$ produce?

   A vertical shift by $k$ units (up if $k > 0$, down if $k < 0$); the horizontal asymptote moves to $y = k$.

9. For $f(x) = b^{x}$, how do $f(x - h)$ and $f(-x)$ transform the graph?

   $f(x - h)$ shifts horizontally by $h$ units (right if $h > 0$); $f(-x) = b^{-x}$ reflects across the $y$-axis.

10. What does the transformation $-f(x)$ do to an exponential graph, and where does its asymptote go?

    $-b^{x}$ reflects the graph across the $x$-axis; the range becomes $(-\infty, 0)$ with asymptote still $y = 0$ (unless combined with a vertical shift).

11. Describe fully the graph of $g(x) = 2^{x-3} + 4$ relative to $f(x) = 2^{x}$.

    Shift right 3 and up 4. Horizontal asymptote $y = 4$; range $(4, \infty)$.

12. State the definition of a logarithm: $\log_{b}(x) = y$ is equivalent to what exponential statement?

    $\log_{b}(x) = y \iff b^{y} = x$ (with $b > 0$, $b \neq 1$, $x > 0$).

13. Why is the logarithmic function $\log_{b}(x)$ described as the inverse of $b^{x}$?

    Because $\log_{b}(b^{x}) = x$ and $b^{\log_{b}(x)} = x$; their graphs are reflections of each other across the line $y = x$.

14. What are the domain and range of $f(x) = \log_{b}(x)$?

    Domain: $(0, \infty)$. Range: $(-\infty, \infty)$ — swapped from the exponential function.

15. What does the common logarithm mean, and how is it written?

    The common logarithm has base 10: $\log(x) = \log_{10}(x)$.

16. What does the natural logarithm mean, and how is it written?

    The natural logarithm has base $e$: $\ln(x) = \log_{e}(x)$.

17. Evaluate $\ln(e)$, $\ln(1)$, and $\log(1)$.

    $\ln(e) = 1$, $\ln(1) = 0$, and $\log(1) = 0$.

18. What is the vertical asymptote of $f(x) = \log_{b}(x)$, and what $x$-intercept does the basic graph have?

    Vertical asymptote $x = 0$ (the $y$-axis); $x$-intercept at $(1, 0)$ since $\log_{b}(1) = 0$.

19. How do the graphs of $\log_{b}(x)$ differ for $b > 1$ versus $0 < b < 1$?

    If $b > 1$ the log graph is increasing; if $0 < b < 1$ it is decreasing.

20. For $g(x) = \log_{2}(x + 5)$, where is the vertical asymptote and what is the domain?

    Vertical asymptote at $x = -5$; domain $(-5, \infty)$ (argument must be positive).

21. State the Product Rule for logarithms.

    $\log_{b}(MN) = \log_{b}(M) + \log_{b}(N)$.

22. State the Quotient Rule for logarithms.

    $\log_{b}\left(\frac{M}{N}\right) = \log_{b}(M) - \log_{b}(N)$.

23. State the Power Rule for logarithms.

    $\log_{b}(M^{p}) = p\,\log_{b}(M)$.

24. What do the identities $\log_{b}(b^{x})$ and $b^{\log_{b}(x)}$ simplify to?

    $\log_{b}(b^{x}) = x$ and $b^{\log_{b}(x)} = x$ (inverse property).

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