To find the equation of the inverse of the function [tex]\( f(x) = \left(\frac{1}{6}\right)^{x-3} \)[/tex], we need to follow a series of steps meticulously:
### Step 1: Express [tex]\( f(x) \)[/tex] as [tex]\( y \)[/tex]
Let's start by writing the function in terms of [tex]\( y \)[/tex]:
[tex]\[ y = \left(\frac{1}{6}\right)^{x-3} \][/tex]
### Step 2: Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]
To find the inverse function, we swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = \left(\frac{1}{6}\right)^{y-3} \][/tex]
### Step 3: Solve for [tex]\( y \)[/tex]
Now, we need to solve this equation for [tex]\( y \)[/tex].
#### Step 3a: Take natural logarithms of both sides
Taking the natural logarithm on both sides:
[tex]\[ \ln(x) = \ln \left(\left(\frac{1}{6}\right)^{y-3}\right) \][/tex]
#### Step 3b: Use the logarithmic power rule
Using the power rule for logarithms [tex]\( \ln(a^b) = b \cdot \ln(a) \)[/tex], we get:
[tex]\[ \ln(x) = (y - 3) \cdot \ln\left(\frac{1}{6}\right) \][/tex]
#### Step 3c: Isolate [tex]\( y \)[/tex]
Next, solve for [tex]\( y \)[/tex]:
[tex]\[ y - 3 = \frac{\ln(x)}{\ln\left(\frac{1}{6}\right)} \][/tex]
Adding 3 to both sides:
[tex]\[ y = \frac{\ln(x)}{\ln\left(\frac{1}{6}\right)} + 3 \][/tex]
### Conclusion
Thus, the inverse function is:
[tex]\[ f^{-1}(x) = \frac{\ln(x)}{\ln\left(\frac{1}{6}\right)} + 3 \][/tex]
So, the equation of the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = \frac{\ln(x)}{\ln\left(\frac{1}{6}\right)} + 3 \][/tex]
