Write an equation for the transformed logarithm shown below, that passes through (0,0) and (-5,2). has a vertical asymptote at x=1

To find the equation of a transformed logarithm that passes through the points (0,0) and (-5,2), and has a vertical asymptote at \( x = 1 \), we can follow these steps:

1. **Start with the basic logarithmic function:**
   The basic form of a logarithmic function is \( y = \log_b(x - h) + k \), where \( h \) is the horizontal shift and \( k \) is the vertical shift.

2. **Determine the horizontal shift:**
   Since the vertical asymptote is at \( x = 1 \), the function is shifted horizontally by 1 unit to the right. Therefore, \( h = 1 \).

   This gives us the function in the form \( y = \log_b(x - 1) + k \).

3. **Find the specific form of the function:**
   To pass through the points (0,0) and (-5,2), we need to determine the base \( b \) and the vertical shift \( k \).

4. **Use the point (0,0):**
   Plugging in (0,0) into the function \( y = \log_b(x - 1) + k \):

   \[
   0 = \log_b(0 - 1) + k
   \]

   Since \( \log_b(-1) \) is undefined, we need to modify the approach. This suggests we need to work with a different structure or an alternative form of transformation.

   Instead, we can consider a vertical and horizontal reflection.

5. **Consider a reflected logarithmic function:**
   A common transformation involving reflection is \( y = -\log_b(1 - x) + k \), where the logarithm is reflected over the y-axis and shifted horizontally and vertically.

   Using the point (0,0):

   \[
   0 = -\log_b(1 - 0) + k
   \]
   \[
   0 = -\log_b(1) + k
   \]
   \[
   \log_b(1) = 0 \implies k = 0
   \]

6. **Use the point (-5,2):**
   Plugging in (-5,2) into the modified function \( y = -\log_b(1 - x) \):

   \[
   2 = -\log_b(1 - (-5))
   \]
   \[
   2 = -\log_b(6)
   \]

   Therefore, \( \log_b(6) = -2 \), and solving for \( b \):

   \[
   b^{-2} = 6 \implies b^2 = \frac{1}{6} \implies b = \frac{1}{\sqrt{6}}
   \]

So, the equation of the transformed logarithmic function is:

\[
y = -\log_{\frac{1}{\sqrt{6}}}(1 - x)
\]

**To summarize:**

The equation of the logarithmic function that passes through the points (0,0) and (-5,2) and has a vertical asymptote at \( x = 1 \) is:

\[
y = -\log_{\frac{1}{\sqrt{6}}}(1 - x)
\]

Would you like more details on the transformations, or do you have any questions?

### Relative Questions
1. How does horizontal shifting affect the graph of a logarithmic function?
2. What are vertical asymptotes, and how are they identified in logarithmic functions?
3. What transformations can be applied to the logarithmic function to achieve reflections?
4. How do the points through which the function passes influence its form?
5. What is the base change formula for logarithms?
6. How does reflecting a function over the y-axis affect its equation?
7. What are the properties of the logarithmic function with different bases?
8. How can logarithmic transformations be used in real-life applications?

### Tip
Remember that transformations of functions (shifts, reflections, and stretches) are a powerful way to manipulate graphs to pass through desired points or have specific asymptotes.

Discover how to find the equation of a transformed logarithmic function passing through the points (0,0) and (-5,2), with a vertical asymptote at x = 1. Learn about logarithmic function transformations and solving for specific parameters. Suitable for advanced high school students.

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