Exponentials & Logs
Logarithms feel like a brand-new, scary thing. They're not: a log is just an exponential read backwards. See the two curves as mirror images and the whole topic clicks into place. For Grade 11.
Undo buttons for each other
The exponential y = bˣ takes an exponent and gives you a value: feed it 3, and with base 2 you get 2³ = 8. The logarithm y = logb x runs it in reverse: feed it 8, and it hands back the exponent, 3. Each one undoes the other — they're inverse functions.
And inverse functions have a beautiful visual signature: their graphs are mirror images across the line y = x. Swap the coordinates of any point on one, and you land on the other.
Drag along the curve 🔵
Drag the blue dot along bˣ. Its green mirror on log_b x is always the same point with its coordinates swapped — reflected across the dashed line y = x. Change the base with the buttons or slider.
What to notice
Swap the coordinates
When the blue point is at (3, 8), the green one is at (8, 3). That's all "inverse" means — the x and y trade places. The dashed line y = x is the mirror they reflect through.
Watch the anchor points
bˣ always passes through (0, 1); its mirror logb x always passes through (1, 0). That's why logb 1 = 0 for every base — you can see it.
Slide the base down toward b < 1 (try ½) and both curves flip: the exponential now decays instead of growing, and the log mirrors it. The special base e ≈ 2.718 gives the "natural" pair eˣ and ln x you'll meet everywhere in Grade 12 and beyond.