What is the inverse function of `f(x)=10^(x)` ?

Answer

To find the inverse function of \( f(x) = 10^x \), we will follow these steps:

### Step 1: Set the function equal to \( y \)
Let \( y = f(x) \). Therefore, we can write:
\[ y = 10^x \]

### Step 2: Solve for \( x \) in terms of \( y \)
To find the inverse, we need to express \( x \) in terms of \( y \). We can do this by taking the logarithm (base 10) of both sides:
\[ \log_{10}(y) = \log_{10}(10^x) \]

### Step 3: Apply the logarithmic identity
Using the property of logarithms that states \( \log_b(a^c) = c \cdot \log_b(a) \), we can simplify the right-hand side:
\[ \log_{10}(y) = x \cdot \log_{10}(10) \]
Since \( \log_{10}(10) = 1 \), this simplifies to:
\[ \log_{10}(y) = x \]

### Step 4: Express the inverse function
Now we have \( x \) in terms of \( y \):
\[ x = \log_{10}(y) \]
Since \( f^{-1}(y) = x \), we can write:
\[ f^{-1}(y) = \log_{10}(y) \]

### Step 5: Replace \( y \) with \( x \)
To express the inverse function in terms of \( x \), we replace \( y \) with \( x \):
\[ f^{-1}(x) = \log_{10}(x) \]

### Final Answer
Thus, the inverse function of \( f(x) = 10^x \) is:
\[ f^{-1}(x) = \log_{10}(x) \]

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