Suppose Marcello invests $500 at 1.2% annually. How long will it take for that amount to double?

Better than investing annually?

e (2.71828…): the natural base

It represents the base rate of growth shared by all continuously

growing processes.

e (2.71828…): the natural base

Q: Why the natural base?A: Because it shows up in

population growth, radioactive decay, and in systems that exhibit

continuous growth or decay.

Suppose we applied a (theoretical) continuous growth rate to an

investment.

Continuously Compounding

Interest Formula:

A = Pert

A = Pert

Note that the expression has

been substituted with

er now.1

lim 1n

ne

n

Take Tacoma’s population (as of 2012) of 202,010 which was growing at

1.8%. If the growth rate remains the same, what will its population be in

2013?Continuously

Compounding Interest Formula:A = Pert

= 202,010e0.018*3

= ~213218

It’s e-asy!

If Portland, with its 2012 population of 603,106, is continuously growing at

1.7% then what will its population be in 2017?

How e-nterestingPortland, with its 2012 population of 603,106, is continuously growing at 1.7%. When what will its population surpass 700,000?

700,000 = 603,106e0.017t [solving for t!]

1.16065… = e0.017t [divide by 603,106]

loge1.16065 = 0.017t [put into log form, and with a base of e]

0.148987… = 0.017t [evaluate dat log]

8.76 396… = t [divide by 0.017]

e-nsane in the Membrane

Portland, with its 2012 population of 603,106, is continuously growing at 1.7%.

When what will its population surpass 800,000?

oh btw

logex = ln(x)

The natural basedeserves a natural logarithm.

So from now on, instead of using loge(3),a shortcut to use is now ln(3).