Logarithmic Function and the Graph of the Inverse
Answer the question about the logarithmic function.
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The logarithmic function y = log₂ x is the inverse of the exponential function y = 2ˣ. The exponential sends 3 to 8, and the logarithm sends 8 back to 3. That is why the coordinates of points swap, and so do the domain and range: the logarithm is defined only for positive x, but it takes all real values. The graph passes through the point (1, 0).
What does this generator do?
On the easy level you determine properties of the function y = logₐ x: domain, range, whether it increases or decreases, a point on the graph and the asymptote. On the medium level you match a formula to a shifted graph of a logarithm. On the hard level you find the inverse of y = a^(x − m) + n, a point on the graph of f⁻¹ and its domain.
How to use it
1) The graph of y = logₐ x passes through the point (1, 0); it increases for a > 1 and decreases for 0 < a < 1. 2) The vertical asymptote is x = m — that is how far the graph is shifted to the right. 3) You find the inverse function by swapping x and y: from y = 2ˣ + 1 you get x = 2ʸ + 1, so y = log₂ (x − 1). 4) The domain of f⁻¹ is the range of f.