Describe the general shape and key features of the graph of a logarithmic function, including any asymptotes.
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The graph of a logarithmic function, such as \(y = \log_b(x)\), typically has a distinct shape that increases slowly and steadily as \(x\) increases. It is defined only for positive values of \(x\), starting from \((1,0)\) where the logarithm of 1 equals 0, then rising steeply for values greater than 1. The key feature is its vertical asymptote at \(x = 0\); as \(x\) approaches zero from the right, the function approaches negative infinity, creating that classic logarithmic curve. Additionally, logarithmic functions are useful in various real-world situations! For instance, they are used in measuring sound intensity in decibels, the Richter scale for earthquake strength, and even in financial calculations like compound interest. The slow growth rate of logarithmic functions makes them perfect for modeling phenomena that increase rapidly at first and then level off over time.