The inverse of the function `y=5^(Inx)` is

Options

A. `x=y^((1)/(In5)),ygt0`
B. `x=y^(In5),ygt0`
C. `x=y^((1)/(In5)),ylt0`
D. `x=5" In "y,ygt0`

Explanation

To find the inverse of the function \( y = 5^{\ln x} \), we will follow these steps:

### Step 1: Rewrite the function
We start with the function:
\[ y = 5^{\ln x} \]

### Step 2: Take the natural logarithm of both sides
To isolate \( x \), we will take the natural logarithm of both sides:
\[ \ln y = \ln(5^{\ln x}) \]

### Step 3: Apply the logarithmic identity
Using the property of logarithms that states \( \ln(a^b) = b \cdot \ln a \), we can simplify the right side:
\[ \ln y = \ln x \cdot \ln 5 \]

### Step 4: Solve for \( \ln x \)
Now, we can isolate \( \ln x \):
\[ \ln x = \frac{\ln y}{\ln 5} \]

### Step 5: Exponentiate both sides to solve for \( x \)
To solve for \( x \), we exponentiate both sides:
\[ x = e^{\frac{\ln y}{\ln 5}} \]

### Step 6: Simplify using properties of exponents
Using the property that \( e^{\ln a} = a \), we can simplify further:
\[ x = y^{\frac{1}{\ln 5}} \]

### Step 7: Write the inverse function
Thus, the inverse function can be expressed as:
\[ y = x^{\ln 5} \]

### Final Result
The inverse of the function \( y = 5^{\ln x} \) is:
\[ x = y^{\frac{1}{\ln 5}} \]