The inverse of the function `log_e` x is

Options

A. `10^x`
B. `10^(-x)`
C. `e^x`
D. `e^(-x)`

Answer

To find the inverse of the function \( f(x) = \log_e x \), we will follow these steps:

### Step 1: Set the function equal to \( y \)
We start by letting \( y = \log_e x \).

### Step 2: Rewrite the equation in exponential form
To find the inverse, we need to express \( x \) in terms of \( y \). The logarithmic equation \( y = \log_e x \) can be rewritten in its exponential form:
\[ x = e^y \]

### Step 3: Define the inverse function
Now that we have \( x \) in terms of \( y \), we can express the inverse function. We replace \( y \) with \( x \) to denote the inverse function:
\[ f^{-1}(x) = e^x \]

### Step 4: Verify the inverse function
To confirm that \( f^{-1}(x) = e^x \) is indeed the inverse of \( f(x) = \log_e x \), we need to check two conditions:
1. \( f(f^{-1}(x)) = x \)
2. \( f^{-1}(f(x)) = x \)

**Check 1:**
\[ f(f^{-1}(x)) = f(e^x) = \log_e(e^x) = x \]

**Check 2:**
\[ f^{-1}(f(x)) = f^{-1}(\log_e x) = e^{\log_e x} = x \]

Since both conditions hold true, we have verified that the inverse function is correct.

### Final Answer:
The inverse of the function \( \log_e x \) is:
\[ f^{-1}(x) = e^x \]

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