If f(x) = 2 ^ log_3(x) , then the value of f ^ - 1 * (16) will be? (a) 27 (b) 1/27 (c) 81 (d) 1/81

To solve for \( f^{-1}(16) \), we need to first understand the function \( f(x) = 2^{\log_3(x)} \).

### Step 1: Find \( f(x) \) in a simpler form
We know:
\[ f(x) = 2^{\log_3(x)} \]

To understand this function, let's rewrite the logarithm using a change of base formula. The change of base formula states:

\[ \log_3(x) = \frac{\log_2(x)}{\log_2(3)} \]

Substituting this into the expression for \( f(x) \):

\[
f(x) = 2^{\log_3(x)} = 2^{\frac{\log_2(x)}{\log_2(3)}}
\]

Now, using the exponent rule \( a^{b \cdot c} = (a^b)^c \), we can rewrite \( f(x) \):

\[
f(x) = \left( 2^{\log_2(x)} \right)^{\frac{1}{\log_2(3)}}
\]

Since \( 2^{\log_2(x)} = x \), we get:

\[
f(x) = x^{\frac{1}{\log_2(3)}}
\]

### Step 2: Find \( f^{-1}(x) \)
To find \( f^{-1}(x) \), we express \( x \) as a function of \( y \):

\[
y = x^{\frac{1}{\log_2(3)}}
\]

Now, solving for \( x \) gives:

\[
y = x^{\frac{1}{\log_2(3)}}
\]
\[
y^{\log_2(3)} = x
\]

So, \( f^{-1}(y) = y^{\log_2(3)} \).

### Step 3: Evaluate \( f^{-1}(16) \)

Now, we substitute \( y = 16 \) into \( f^{-1}(y) \):

\[
f^{-1}(16) = 16^{\log_2(3)}
\]

Since \( 16 = 2^4 \):

\[
f^{-1}(16) = (2^4)^{\log_2(3)}
\]
\[
= 2^{4 \cdot \log_2(3)}
\]
\[
= 2^{\log_2(3^4)}
\]
\[
= 3^4
\]
\[
= 81
\]

Therefore, the value of \( f^{-1}(16) \) is \(\boxed{81}\).

### Answer:
(c) 81

Would you like more details, or do you have any questions? 

Here are five related questions to expand on this topic:

1. What is the domain and range of the function \( f(x) = 2^{\log_3(x)} \)?
2. How would you find the inverse of a logarithmic function in general?
3. Can you express \( f(x) = 2^{\log_3(x)} \) using natural logarithms?
4. What is the graphical interpretation of finding the inverse of a function?
5. How would you solve for \( f^{-1}(32) \) if \( f(x) = 2^{\log_3(x)} \)?

**Tip:** Understanding the change of base formula for logarithms can simplify many calculations and is a valuable tool when dealing with functions involving different bases.

Discover how to find the inverse function value f^-1(16) for f(x) = 2^(log_3(x)). Follow a detailed step-by-step solution involving exponential functions, logarithms, and inverse functions. Suitable for high school students learning advanced algebra and functions.

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