Exponential function y = aˣ and logarithmic function y = logₐx
MathematicsFunctions & GraphsAges 16–17
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Functions & Graphs
The graphs of y = aˣ and y = logₐx are reflections of each other across the line y = x. Drag point M on the exponential graph and its mirror point M' moves on the logarithmic graph; drag the a-slider to see the function increasing for a > 1 and decreasing for 0 < a < 1. The Log laws screen shows a logarithm as an exponent on a powers-of-b scale, the product and quotient laws on a slide rule, the power law, change of base and b^(x+k) = b^k·b^x as graph transformations; the Equations screen solves A·b^(ct) = d graphically and with logarithms, including doubling time and half-life. The Base e screen shows why e is special: only for a = e does the gradient of aˣ equal its value; the gradient of e^(kx) is k·e^(kx); ln x is the area under y = 1/t and the reflection of eˣ in y = x; and (1 + 1/n)ⁿ → e links to continuous compounding.
Lesson: Exponential functions, logarithmic functions
What it shows
The exponential function y = aˣ and the logarithm y = logₐx are inverses, so their graphs are reflections in y = x. A logarithm is an exponent; the laws log(xy) = log x + log y, log(x/y) = log x − log y and log(xᵏ) = k·log x follow from the index laws, and taking logs solves A·b^(ct) = d. The number e ≈ 2.718 is the only base for which the gradient of aˣ equals its value, so the gradient of e^(kx) is k·e^(kx). The natural logarithm ln x is the area under y = 1/t from 1 to x, and (1 + 1/n)ⁿ tends to e.
How to use
On Graphs, drag the Base a slider across 1 and drag point M along y = aˣ. On Log laws, choose a Law and a Base, then drag on the picture. On Equations, set A, b, c and d or choose a Problem and tick Logarithmic y-axis. On Base e, choose a View, then change a or press a = e, change k, or set n and r, and drag on the graph.
Parameters you can change
- Base a (a > 0, a ≠ 1) 0.1–5
- Starting x-coordinate of point M -3–3
- Application shown Graph only, Population growth (×a per period), Radioactive decay (a of the amount remains per period, a < 1)
- Show graph y = logₐx
- Show line y = x and the pair of mirror points
- Screen Graphs, Log laws, Equations, Base e
- Law of logarithms A log is an exponent, Product law: log(xy), Quotient law: log(x/y), Power law: log(xᵏ), log(kx): translation, Change of base, bˣ⁺ᵏ = bᵏ·bˣ
- Base b (laws) 2, 3, 10, e
- Value x (laws) 0.1–100
- Value y (slide rule) 0.1–10
- Parameter k -3–10
- New base c 2, 3, 10, e
- Starting value A 1–1000
- Base b (equation) 0.5–3
- Constant c 0.1–12
- Target value d 1–5000
- Problem Choose d, Doubling time (d = 2A), Half-life (d = A/2)
- Logarithmic y-axis
- Base e: view Gradient of aˣ, Gradient of e^(kx), ln x as an area, (1 + 1/n)ⁿ → e
- Base a (Base e screen) 1.2–4
- Constant k in y = e^(kx) -2–2
- x-coordinate of the tangent point -3–2
- Upper limit x of the area under y = 1/t 0.1–8
- Number of compounding periods n 1–1000000
- Annual interest rate r 1–100 %
Questions to explore
- Using the reflection of y = aˣ in the line y = x, why is logₐx defined only for x > 0?
- If M(2; 4) lies on y = 2ˣ, which point must lie on y = log₂x?
- A population grows by a factor of 1.1 each period. After how many periods has it doubled?
