(This is the opposite from exponential functions, where a larger base meant a steeper graph.) A larger base also makes the graph closer to the x-axis for y > 0 (or x > 1) and closer to the y-axis for y < 0 (or x < 1). Notice that a larger base makes the graph less steep. Let’s compare the logarithmic graphs you’ve seen: f( x) = log 2 x, f( x) = log 3 x, and f( x) = log 4 x. Remember that logarithmic functions get close to the y-axis (but won’t touch or cross it). Connect the points as best you can, using a smooth curve (not a series of straight lines). Since the points are not on a line, you can’t use a straightedge. Because you know the square root of 4, try x =. In this case, (16, 2) won’t show on the graph. Use the table as ordered pairs and plot the points. Start with a table of values, choosing the y values and calculating x. Log b x = y if and only if b y = x, where x > 0 and b > 0, b ≠ 1. The logarithm of x in base b is written log b x and is defined as: The range of the logarithmic function (in blue) is all real numbers. Similarly, the domain of the exponential function (in red) is all real numbers. Since the input and output have been switched, the domain ( x values) of the logarithmic function (in blue) is positive real numbers. You can see from the graph that the range ( y values) of the exponential function (in red) is positive real numbers. In general, y = log b x is read, “ y equals log to the base b of x,” or more simply, “ y equals log base b of x.” As with exponential functions, b > 0 and b ≠ 1. The logarithmic function for x = 2 y is written as y = log 2 x or f( x) = log 2 x. The equation x = 2 y is often written as a logarithmic function (called log function for short). (This makes sense, because y in the first table becomes x in the second table, and vice versa.) Another way to put it, if you rotate the red curve about the line y = x, it will coincide with the blue curve. As shown in the graph, the two curves are symmetrical about the line y = x. The graphs of these two relationships should have the same general shape. Note the two tables are the same except the columns are reversed-the point (1, 2) taken from the first table will be the point (2, 1) in the second table. Consider these tables of values using a base of 2.
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