What is the inverse of the function `y=5^(logx)`?

Options

A. `x=5^(1//logy)`
B. `x=y^(1//log5)`
C. `x=5^(logy)`
D. `x=y^(log5)`

Explanation

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

### Step 1: Write the original function
We start with the function:
\[ y = 5^{\log x} \]

### Step 2: Swap \( x \) and \( y \)
To find the inverse, we swap \( x \) and \( y \):
\[ x = 5^{\log y} \]

### Step 3: Take the logarithm of both sides
Next, we take the logarithm of both sides:
\[ \log x = \log(5^{\log y}) \]

### Step 4: Apply the logarithmic identity
Using the property of logarithms that states \( \log(a^b) = b \cdot \log a \), we can simplify the right-hand side:
\[ \log x = \log y \cdot \log 5 \]

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

### Step 6: Exponentiate to solve for \( y \)
To solve for \( y \), we exponentiate both sides:
\[ y = 10^{\frac{\log x}{\log 5}} \]

### Step 7: Simplify using properties of exponents
Using the property that \( a^{\log_b c} = c^{\log_b a} \), we can rewrite this as:
\[ y = x^{\frac{1}{\log 5}} \]

### Final Step: Write the inverse function
Thus, the inverse function is:
\[ y = x^{\frac{1}{\log 5}} \]

### Conclusion
The inverse of the function \( y = 5^{\log x} \) is:
\[ f^{-1}(x) = x^{\frac{1}{\log 5}} \]