Question

which is the inverse of f(x)=log(under)1/3 x

Answer 1

To find the inverse of the function \( f(x) = \log_{1/3}(x) \), we need to follow a few steps:

1. **Replace \( f(x) \) with \( y \)**:
\[
y = \log_{1/3}(x)
\]

2. **Express the logarithmic equation in exponential form**:
The logarithmic equation \( y = \log_{b}(x) \) is equivalent to \( x = b^y \). For our function:
\[
x = \left(\frac{1}{3}\right)^y
\]

3. **Solve for \( y \)**:
To find the inverse, we swap \( x \) and \( y \):
\[
y = \left(\frac{1}{3}\right)^x
\]

4. **Replace \( y \) with \( f^{-1}(x) \)**:
Therefore, the inverse function is:
\[
f^{-1}(x) = \left(\frac{1}{3}\right)^x
\]

So, the inverse of the function \( f(x) = \log_{1/3}(x) \) is:
\[
f^{-1}(x) = \left(\frac{1}{3}\right)^x
\]