The value of `16^("log"4^(3))`, is

Options

A. 8
B. 3
C. 4
D. 9

Explanation

To solve the expression \( 16^{\log_4(3)} \), we can follow these steps:

### Step 1: Rewrite the base
First, we rewrite \( 16 \) in terms of base \( 4 \):
\[ 16 = 4^2 \]
So, we can express the original expression as:
\[ 16^{\log_4(3)} = (4^2)^{\log_4(3)} \]

### Step 2: Apply the power of a power property
Using the property of exponents, \( (a^m)^n = a^{m \cdot n} \), we can simplify the expression:
\[ (4^2)^{\log_4(3)} = 4^{2 \cdot \log_4(3)} \]

### Step 3: Use the logarithmic identity
Now, we can use the logarithmic identity which states that \( a \cdot \log_b(c) = \log_b(c^a) \). Thus, we can rewrite \( 2 \cdot \log_4(3) \) as:
\[ 2 \cdot \log_4(3) = \log_4(3^2) \]
This gives us:
\[ 4^{2 \cdot \log_4(3)} = 4^{\log_4(3^2)} \]

### Step 4: Apply the property of logarithms
Now, we can use the property \( b^{\log_b(x)} = x \):
\[ 4^{\log_4(3^2)} = 3^2 \]

### Step 5: Calculate the final value
Calculating \( 3^2 \):
\[ 3^2 = 9 \]

Thus, the value of \( 16^{\log_4(3)} \) is:
\[ \boxed{9} \]

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