High School: Functions
Building Functions HSF-BF.B.5
5. Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Students should have pieced together the puzzle by now, more or less. The inverse of adding is subtracting. The inverse of multiplying is dividing. The inverse of squaring is square-rooting (and the same goes for any circle-roots or triangle-roots). Now it's time for the final piece of the puzzle: the inverse of exponentiating is logarithmifying (which may or may not be a real word).
Students should already know that an exponential equation a = bc can be rewritten in logarithmic form as c = logba or
They may not have used logarithms in a while, but it should still be there, under the piles of memorized Jonas Brothers lyrics and Gangnam Style dance moves. After they've yanked it back out and dusted it off, tell them why it's useful.
Students should be able to use this exponent-logarithm relationship when finding the inverse of a function. For instance, we have the function f(x) = 3x, which we can treat as y = 3x. Finding the inverse means switching x and y, and then solving for y. So what we have is really x = 3y. We can now use the wonderful world of logarithms to solve for y. We should get y = log3x ≈ 2.1logx.
If your students aren't convinced that these functions are inverses, do whatever you need to do to prove it. You can graph them and show the line of symmetry, plot points and switch them, or calculate f(f-1(x)) and prove that it equals x. All of those should be enough evidence to support the fact that exponentials and logarithms are inverses.