find the inverse of log of base 1.2 of (x-6)

To find the inverse of \( \log_{1.2}(x-6) \), follow these steps:

### 1. Express the logarithmic function as an equation:
Let \( y = \log_{1.2}(x - 6) \). This means:
\[
y = \log_{1.2}(x - 6)
\]
This is equivalent to:
\[
1.2^y = x - 6
\]

### 2. Solve for \( x \):
Now, solve for \( x \) in terms of \( y \):
\[
x = 1.2^y + 6
\]

### 3. Express the inverse function:
The inverse function will swap \( x \) and \( y \). So, replace \( y \) with \( x \), and \( x \) with \( y \) to get the inverse function:
\[
f^{-1}(x) = 1.2^x + 6
\]

### Final answer:
The inverse of \( \log_{1.2}(x - 6) \) is:
\[
f^{-1}(x) = 1.2^x + 6
\]

Let me know if you'd like any further details or clarifications! 

### Here are 5 follow-up questions you might find useful:
1. How do you find the inverse of other logarithmic functions?
2. What does the graph of \( f(x) = \log_{1.2}(x - 6) \) look like?
3. How can we graph the inverse function \( f^{-1}(x) = 1.2^x + 6 \)?
4. What is the domain and range of \( \log_{1.2}(x - 6) \) and its inverse?
5. How do logarithmic and exponential functions relate to each other in general?

### Tip:
Remember that the inverse of a logarithmic function is always an exponential function!

Learn how to find the inverse of the logarithmic function log base 1.2 of (x-6). This detailed guide shows the steps to express the function as an equation, solve for x, and derive the inverse. Suitable for high school students in Grades 10-12.

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