Verified Answer :

  • Understand that \(\log_2 8 = y\) is equivalent to the exponential form \(2^y = 8\).
  • Express 8 as a power of 2, which is \(2^3\).
  • Equate the exponents to find \(y = 3\).
  • Conclude that \(\log_2 8 = \boxed{3}\).

Explanation

  1. Understanding Logarithms

The goal is to calculate the value of the logarithmic expression \(\log_2 8\). A logarithm \(\log_b x\) represents the exponent to which the base \(b\) must be raised to equal the argument \(x\). In this case, the base \(b\) is 2 and the argument \(x\) is 8. So, we are looking for a value \(y\) such that \(2^y = 8\).

  1. Solving the Exponential Equation

By definition, \(\log_2 8 = y\) is equivalent to the exponential equation \(2^y = 8\). We need to express 8 as a power of 2. We can list powers of 2:
\(2^1 = 2\)
\(2^2 = 2 \times 2 = 4\)
\(2^3 = 2 \times 2 \times 2 = 8\)
Since \(8 = 2^3\), we can substitute this into our equation: \(2^y = 2^3\).

  1. Final Result

Since the bases are equal (both are 2), the exponents must be equal for the equation to hold true. Therefore, \(y = 3\). This confirms that \(\log_2 8 = 3\).

Examples

Logarithms are used in various fields to scale data that spans several orders of magnitude. For example, in chemistry, the pH scale uses logarithmic values to represent the concentration of hydrogen ions in a solution. A solution with a pH of 3 is ten times more acidic than a solution with a pH of 4, demonstrating how logarithms help simplify the measurement of values that would otherwise be inconvenient to express on a linear scale. Similarly, in sound engineering, the decibel (dB) scale uses a logarithmic basis to represent sound intensity, making it easier to manage the wide range of sound pressure levels humans can perceive.

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