Exponential

Add nu­mer­ous other sum­maries. See, e.g., the sum­maries at /​p/​log­a­r­ithm. I sug­gest mak­ing the main page for ex­po­nen­tial mir­ror the main page at /​p/​log­a­r­ithm.

An ex­po­nen­tial is a func­tion that can be rep­re­sented by some con­stant taken to the power of a vari­able. The name comes from the fact that the vari­able is the ex­po­nent of the ex­pres­sion.

Ex­po­nen­tial Growth

Ex­po­nen­tials are most use­ful in de­scribing growth pat­terns where the growth rate is pro­por­tional to the amount of the thing that’s grow­ing. They can be rep­re­sented by the for­mula: \(f(x) = c \times a^x\), where \(c\) is the start­ing value and \(a\) is the growth fac­tor.

The clas­sic ex­am­ple of ex­po­nen­tial growth is com­pound in­ter­est. If you have $100 in a bank ac­count that gives you 2% in­ter­est ev­ery year, then ev­ery year your money is mul­ti­plied by \(1.02\). This means you can rep­re­sent your ac­count bal­ance as \(f(x) = 100 \times 1.02^x\), where \(x\) is the num­ber of years your money has been in the ac­count.

Another ex­am­ple is a di­vid­ing cell. If one cell is placed into an in­finite cul­ture and splits once ev­ery hour, the num­ber of cells in the cul­ture af­ter \(x\) hours is \(f(x) = 1 \times 2^x\) (as­sum­ing none of the cells die).

Re­cur­sive definition

We men­tioned ear­lier that in an ex­po­nen­tial func­tion, the growth rate is pro­por­tional to the amount of the thing that’s grow­ing. In the com­pound in­ter­est ex­am­ple, we can write each value in terms of the pre­vi­ous value: \(f(x) = f(x-1) \times 1.02\). There­fore, the amount of growth at ev­ery step can be rep­re­sented as \(\Delta f(x) = f(x+1) - f(x) = 0.02 \times f(x)\).

This makes ex­po­nen­tial growth a mem­o­ryless growth func­tion, as the growth rate de­pends only on cur­rent in­for­ma­tion. Com­pare this to sim­ple in­ter­est, where the in­ter­est only grows at a per­centage of the ini­tial value. Then we would have to write: \(f(x) = f(x-1) + 0.02 \times f(0)\), and be­cause \(f(0)\) is con­stant while \(f(x)\) con­tinues to grow, we can­not ex­press the growth rate in terms of only cur­rent in­for­ma­tion — we have to keep “in mem­ory” the ini­tial bal­ance to calcu­late the growth rate.

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