Evaluate: $\log _{12} 12$
The problem asks to evaluate the logarithmic expression \(\log_{12} 12\). The definition of a logarithm states that for a base \(b\) and an argument \(x\), the value \(y = \log_{b} x\) is the exponent to which the base must be raised to equal the argument. That is, \(b^y = x\). In this problem, the base \(b = 12\) and the argument \(x = 12\). We seek the value of \(y\) such that \(12^y = 12\).
A fundamental property of logarithms is that for any base \(b > 0\) such that \(b eq 1\), the logarithm of the base itself is equal to 1: \(\log_{b} b = 1\). Applying this property directly to \(\log_{12} 12\), we can immediately see that the base 12 raised to the power of 1 results in 12, because \(12^1 = 12\). Thus, \(\log_{12} 12 = 1\).
Comparing our result to the given options:
The correct choice is \(1\). Therefore, the value of the expression is \(\boxed{1}\).
Logarithms are used in various fields, such as measuring the magnitude of earthquakes using the Richter scale or calculating the acidity (pH level) of a chemical solution. For example, if you have a solution with a hydrogen ion concentration of \(10^{-7}\) moles per liter, you use the common logarithm (\(\log_{10}\)) to determine the pH: \(\text{pH} = -\log_{10}(10^{-7}) = 7\). Understanding the basic properties of logarithms, such as \(\log_b b = 1\), serves as a foundation for solving these more complex scientific equations.