![]() ![]() The range of f is the interval (- ∞, + ∞).ī - The vertical asymptote is obtained by solving |x | = 0Īs x approaches 0 from the right (x > 0), f(x) decreases without bound. Hence, the domain is the set of all real numbers except 0. ![]() Join the points by a smooth curve and f increases as x approaches 4 from the right.Ī - The domain of f is the set of all x values such that | x | > 0. Let us now sketch all the points and the vertical asymptote. We need extra points to be able to graph f.į(4.5) = -3ln(4.5 - 4) approximately equal to 2.08į(8) = -3ln(8 - 4) approximately equal to - 4.16į(14) = -3 ln(14 - 4) approximately equal to - 6.91 f(0) is undefined since x = 0 is not a value in the domain of f.ĭ - So far we have the domain, range, x intercept and the vertical asymptote. How do we know this?į(4.001) = -3 ln(0.001) which is approximately equal to 20.72į(4.000001) = - 3 ln(0.000001) which is approximately equal to 41.45Ĭ - To find the x intercept we need to solve the equation f(x) = 0 or - 3 ln (x - 4) = 0 Solve the above inequality to obtain the domainī - The vertical asymptote is obtained by solving the equationĪs x approaches 4 from the right (x > 4), f(x) increases without bound. The set of all x values such that x - 4 > 0 ![]() We now join the different points by a smooth curve. The graph never cuts the vertical asymptote. Close to the vertical asymptote x = -2, the graph of f decreases without bound as x approaches -2 from #LOGARITHMIC GRAPH HOW TO#We now have more information on how to graph f. Let us consider a point at x = -3/2 (half way between the x intercept and the vertical asymptote) and another point at x = 2.į(-3/2) = log 2 (-3/2 + 2) = log 2 (1/2) = log 2 (2 -1 ) = -1. The y intercept is at the point (0, f(0)) = (0, log 2 (0 + 2)) = (0, 1).ĭ - So far we have the domain, range, x and y intercepts and the vertical asymptote. Rewrite the above equation in exponential form Let us take calculate values of f as x approaches - 2 from the right (x > - 2).į(-1.5) = log 2 (-1.5 + 2) = log 2 (1/2) = -1į(-1.99) = log 2 (-1.99 + 2) = log 2 (0.01) which is approximately equal to -6.64į(-1.999999) = log 2 (-1.999999 + 2) = log 2 (0.000001) which is approximately equal to -19.93.Īs we continue with values of x closer to -2, f(x) decreases without bound.Ĭ - To find the x - intercept we need to solve the equation f(x) = 0 or log 2 (x + 2) = 0 The range of f is given by the interval (- ∞, + ∞).ī - The vertical asymptote is obtained by solving the equation: x + 2 = 0Īs x approaches -2 from the right (x > -2), f(x) decreases without bound because there is a vertical asymptote. Solve the above inequality to obtain the domain: x > - 2 Find the x and y intercepts of the graph of f if there are any.Ī - The domain of f is the set of all x values such that x + 2 > 0.Find the vertical asymptote of the graph of f.You may want to review all the above properties of the logarithmic function interactively. This function has an x intercept at (1, 0) and f increases as x increases. The range of f is given by the interval (- ∞, + ∞).įunction f has a vertical asymptote given by the vertical line x = 0. The domain of function f is the interval (0, + ∞). We first start with the properties of the graph of the basic logarithmic function of base a,į (x) = log a (x), a > 0 and a not equal to 1. Review Properties of Logarithmic Functions The properties such as domain, range, vertical asymptotes and intercepts of the graphs of these functions are also examined in details. Graphing and sketching logarithmic functions: a step by step tutorial. ![]()
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