Graphing Natural Log Functions (ln x) Examples for Pre-Calculus

📚 Quick Study Guide

• 🌱 The natural logarithm, denoted as $ln(x)$, is the logarithm to the base $e$ (Euler's number, approximately 2.71828).
• 📈 The graph of $y = ln(x)$ has a vertical asymptote at $x = 0$.
• 🧭 The domain of $y = ln(x)$ is all positive real numbers, i.e., $x > 0$.
• 🔑 The range of $y = ln(x)$ is all real numbers.
• ✂️ The graph of $y = ln(x)$ passes through the point $(1, 0)$ because $ln(1) = 0$.
• 💡 Transformations of $y = ln(x)$: $y = a \cdot ln(x - h) + k$ involves vertical stretch/compression (a), horizontal shift (h), and vertical shift (k).
• 📝 To graph, identify the vertical asymptote, plot key points, and sketch the curve.

Practice Quiz

1. What is the domain of the function $y = ln(x + 3)$?
   1. $x > 0$
   2. $x > -3$
   3. $x < 3$
   4. All real numbers


2. Which of the following is true about the graph of $y = ln(x)$?
   1. It has a horizontal asymptote at $y = 0$.
   2. It passes through the point $(0, 1)$.
   3. It has a vertical asymptote at $x = 0$.
   4. It has a domain of all real numbers.


3. What is the vertical asymptote of the function $y = ln(x - 2)$?
   1. $x = 0$
   2. $x = -2$
   3. $x = 2$
   4. $y = 2$


4. How does the graph of $y = ln(x) + 1$ compare to the graph of $y = ln(x)$?
   1. Shifted 1 unit to the left.
   2. Shifted 1 unit to the right.
   3. Shifted 1 unit up.
   4. Shifted 1 unit down.


5. Which point lies on the graph of $y = ln(e)$?
   1. (0, 1)
   2. (e, 0)
   3. (e, 1)
   4. (1, e)


6. What is the range of the function $y = 2ln(x)$?
   1. $y > 0$
   2. $y < 0$
   3. All real numbers
   4. $y > 2$


7. What transformation is applied to $y=ln(x)$ to obtain $y=ln(3x)$?
   1. Vertical stretch by a factor of 3
   2. Horizontal stretch by a factor of 3
   3. Horizontal compression by a factor of 1/3
   4. Vertical compression by a factor of 1/3