Summary
Highlights
Logarithmic functions are the inverse of exponential functions, meaning they have a vertical asymptote instead of a horizontal one. They increase at a decreasing rate. The four basic shapes depend on the signs of log x and x. For log x, the graph goes towards quadrant one. For log(-x), it reflects across the y-axis towards quadrant two. For -log(x), it reflects across the x-axis towards quadrant four. For -log(-x), it reflects across the origin towards quadrant three.
To graph log base 2 of x, set the inside (x) equal to zero for the vertical asymptote (x=0). Then, set x equal to one and the base (2), which gives the points (1, 0) and (2, 1). Plot these points and draw the curve starting from the vertical asymptote and passing through the points. The range for logarithmic functions is all real numbers, and the domain is from 0 to infinity, which can be found by setting the inside greater than zero.
For log base 3 of (x - 1) + 2, set (x - 1) to zero for the vertical asymptote (x=1). Then, set (x - 1) equal to one and the base (3). This yields x=2 and x=4. Calculate the corresponding y-values: when x=2, y=2; when x=4, y=3. Plot the vertical asymptote at x=1 and the points (2,2) and (4,3). Draw the graph starting from the asymptote and moving through the points. The range is all real numbers, and the domain is (1, infinity).
To graph 3 - log base 4 of (5 - x), set (5 - x) to zero for the vertical asymptote (x=5). Set (5 - x) to one and the base (4) to find other x-values, which are x=4 and x=1. Calculate the y-values: when x=1, y=2; when x=4, y=3. Plot the vertical asymptote at x=5 and the points (1,2) and (4,3). The graph starts from the asymptote and follows the points. The negative sign in front of x indicates movement to the left, and the negative sign in front of the log indicates movement downwards, towards quadrant three relative to the starting point.