A logarithmic function is a type of mathematical function that models relationships between variables where one variable changes at a rate that is proportional to the logarithm of the other variable. The logarithm of a number is the exponent to which another fixed value, known as the base, must be raised to produce that number.
To graph a logarithmic function, you first need to identify the base of the logarithm. Then, create a table of values by choosing different input values and using the logarithm function to find their corresponding output values. Plot these points on a graph and connect them with a smooth curve. You may also use a calculator or graphing software to create a more accurate graph.
The domain of a logarithmic function is all positive real numbers, as the logarithm of a negative number is undefined. The range of a logarithmic function is all real numbers, as the output values can be positive or negative depending on the base of the logarithm.
The key characteristics of a logarithmic function graph include a vertical asymptote at x = 0, a curve that gets closer and closer to the x-axis but never touches it, and a horizontal asymptote at y = 0. The graph is also reflected over the line y = x, and the domain is all positive real numbers.
To transform a logarithmic function graph, you can use the properties of logarithms, such as the power rule, product rule, and quotient rule. These rules allow you to change the base of the logarithm, factor out constants, and combine logarithmic expressions. You can also use vertical and horizontal translations, reflections, and stretches to transform the graph.