Understanding the Logarithm Problem

The problem asks us to find the value of the expression \(16^{\log_2 3}\). This involves understanding how exponents and logarithms work together, specifically how they relate to each other through their properties.

Key Logarithm Properties Used

To solve this expression involving logarithms and exponents, we will utilize a few fundamental properties:

• Property 1 (Exponent Rule): \((a^m)^n = a^{mn}\)
  This rule states that when a power is raised to another power, you multiply the exponents together.
• Property 2 (Logarithm Rule): \(n \log_a b = \log_a b^n\)
  This property allows us to move a coefficient (a number multiplying the logarithm) into the logarithm's argument as an exponent.
• Property 3 (Fundamental Logarithm Identity): \(a^{\log_a b} = b\)
  This is a very important identity. It shows that if the base of an exponential expression is the same as the base of the logarithm in its exponent, the result is simply the argument of the logarithm.

Step-by-Step Calculation of 16log2 3

Let's break down the calculation for \(16^{\log_2 3}\) into clear, sequential steps:

1. Step 1: Express the base of the exponent in terms of the logarithm's base.
   The base of our exponent is 16, and the base of our logarithm is 2. We can rewrite 16 as a power of 2:
   \[16 = 2 \times 2 \times 2 \times 2 = 2^4\]
   So, the original expression \(16^{\log_2 3}\) transforms into:
   \[(2^4)^{\log_2 3}\]
2. Step 2: Apply the Exponent Rule.
   Using Property 1, \((a^m)^n = a^{mn}\), we multiply the exponents:
   \[(2^4)^{\log_2 3} = 2^{4 \cdot \log_2 3}\]
3. Step 3: Apply the Logarithm Rule.
   Using Property 2, \(n \log_a b = \log_a b^n\), we move the coefficient 4 (which is multiplying the logarithm) into the logarithm's argument as a power:
   \[2^{4 \cdot \log_2 3} = 2^{\log_2 3^4}\]
4. Step 4: Evaluate the power inside the logarithm.
   Now, calculate the value of \(3^4\):
   \[3^4 = 3 \times 3 \times 3 \times 3 = 9 \times 9 = 81\]
   Substitute this value back into the expression:
   \[2^{\log_2 81}\]
5. Step 5: Apply the Fundamental Logarithm Identity.
   Finally, using Property 3, \(a^{\log_a b} = b\), since the base of the exponent (2) matches the base of the logarithm (2), the entire expression simplifies to the argument of the logarithm:
   \[2^{\log_2 81} = 81\]

Final Value of 16log2 3

After applying these fundamental logarithm and exponent rules step-by-step, the value of the expression \(16^{\log_2 3}\) is 81.

This result shows how understanding logarithm properties simplifies complex exponential expressions.