Find the exact value of the logarithm without using a calculator.
$\log _2 \sqrt{2}$
The goal is to calculate the value of the expression \(\log_{2} \sqrt{2}\). The expression consists of a logarithm with base 2 and an argument of \(\sqrt{2}\). Recall that \(\sqrt{x}\) can be written as an exponent with a power of \(\frac{1}{2}\), specifically \(\sqrt{2} = 2^{1/2}\). This allows us to express the argument of the logarithm as a power of the base, which is a common technique for solving logarithmic expressions.
We substitute the exponential form of the square root into the logarithm:
\[\log_{2} \sqrt{2} = \log_{2} (2^{1/2})\]
Using the logarithmic property \(\log_b(b^k) = k\), where \(b = 2\) and \(k = \frac{1}{2}\), we can directly simplify this expression. Alternatively, using the power rule for logarithms \(\log_b(x^k) = k \log_b(x)\), we have:
\[\log_{2} (2^{1/2}) = \frac{1}{2} \log_{2} (2)\]
Since the logarithm of the base to itself is always equal to 1, \(\log_{2} (2) = 1\), the expression simplifies to \(\frac{1}{2} \times 1 = \frac{1}{2}\).
We have successfully determined the value of \(\log_{2} \sqrt{2}\) by converting the square root to an exponent and applying the fundamental properties of logarithms. The exact value is \(\frac{1}{2}\).
Logarithms are used in many scientific fields to compress large ranges of values into a more manageable scale. For instance, the Richter scale for measuring earthquake magnitude is logarithmic, meaning that a magnitude 6 earthquake is 10 times more powerful than a magnitude 5. Understanding the base and the properties of logarithms, such as how \(\log_{2} \sqrt{2} = 1/2\) works, is foundational for grasping these scales and for advanced calculations in computer science, acoustics, and data analysis.