Combining Logarithms: A Step-by-Step Solution with Laws of Logarithms

Combining the Logarithmic Expression log₂(7) - 7 log₂(x) + 1/2 log₂(x + 1)

This problem involves combining a logarithmic expression into a single logarithm. To achieve this, we'll utilize the laws of logarithms, specifically the rules of logarithmic addition and subtraction.

Here's a step-by-step solution:

1. Apply the Power Rule: The coefficients of the logarithms can be moved to the exponents of their respective arguments:

   log₂(7) - 7 log₂(x) + 1/2 log₂(x + 1) = log₂(7) + log₂(x^(-7)) + log₂((x + 1)^(1/2))
   

2. Apply the Product Rule: Logarithms with addition between them can be combined into a single logarithm where the arguments are multiplied:

   log₂(7) + log₂(x^(-7)) + log₂((x + 1)^(1/2)) = log₂(7 * x^(-7) * (x + 1)^(1/2))
   

3. Simplify: Combine the terms within the logarithm:

   log₂(7 * x^(-7) * (x + 1)^(1/2)) = log₂(7(x + 1)^(1/2) / x^7)
   

Therefore, the expression log₂(7) - 7 log₂(x) + 1/2 log₂(x + 1) can be combined into the single logarithm log₂(7(x + 1)^(1/2) / x^7).

This example demonstrates how the laws of logarithms provide a powerful tool for simplifying complex logarithmic expressions.