What is the value of $\log _4 16$ ?
The problem asks for the value of the logarithm \(\log_4 16\). A logarithm \(\log_b x\) is defined as the exponent \(y\) to which a base \(b\) must be raised to produce the number \(x\). In our case, the base \(b = 4\) and the argument \(x = 16\). We want to find the exponent \(y\) such that \(4^y = 16\).
We set up the equation \(\log_4 16 = y\). By the definition of logarithms, this is equivalent to the exponential equation:
\[4^y = 16.\]
To solve for \(y\), we express the number \(16\) as a power of \(4\). We know that \(4^1 = 4\) and \(4^2 = 4 \times 4 = 16\). Therefore, we can substitute \(16\) with \(4^2\):
\[4^y = 4^2.\]
Since the bases are the same, we can equate the exponents:
\[y = 2.\]
We have determined that \(\log_4 16 = 2\). Comparing this result with the given options (1, 2, 4, 8), we confirm that the correct value is 2. Thus, \(\log_4 16 = \boxed{2}\).
Logarithms are used in various fields, such as measuring the sound intensity in decibels or determining the Richter scale magnitude for earthquakes. For instance, if an earthquake's intensity is 100 times stronger than a reference intensity, the Richter scale reading involves the base-10 logarithm: \(\log_{10} 100 = 2\). Understanding how to simplify logarithms like \(\log_4 16 = 2\) provides a foundation for working with these exponential scales in science and finance.