Exponential Concepts

We will look at a few related things

  • logarithmic scales
  • exponential functions
  • exponential notation

Each of these provide tools for working with situations that exhibit exponential growth. For example, a population or bank balance is multiplied by a number each time a year goes by.

Exponential models fit these things well:

  • Populations
  • Contagious disease spread
  • Credit card balances
  • Viral videos
  • Piano Keyboard

Mathematically, these functions have a fixed ratio between the y-axis values when the corresponding points on the x-axis have the same difference.

Exponential Examples

Folding paper

  • Zero folds 2^0 makes one sheet thick
  • One fold 2^1 is two sheets thick
  • How many times can you fold?
  • How can we express the number of pages by the number of folds?
  • Can we write out the pattern?

\textrm{pages} = 2^{\textrm{folds}}

Rabbits

  • start with two
  • wait one year
  • now we have double (4)
  • wait another year
  • now we have eight (8)
  • how many in 5 years?

The number of rabbits at the start of the generation (starting with generation zero) is \textrm{rabbits} = 2^{\textrm{generations+1}}

Money grows the same way

  • Start with $1000
  • Grow by 10%
  • Now $1100
  • Grow by 10%
  • Now $1210

1000 \cdot (1+0.10)^{years}

Piano Scale

Exponential Growth

Constant Growth

This is in contrast to linear growth where if you wait for the amount to increase by a fixed amount, the amount will always increase by that fixed amount in that amount of time.

For linear growth, the slope of the function is a number that does not change along the function.

Exponential notation

You’ve probably seen notation like x^2 or x^3 several times before. In exponential notation, our symbol for a variable, x is in the exponent.

For example 2^x or 3^x.

Recall that 1 = 2^0, 2 = 2^1, 2 \cdot 2 = 2^2 = 4, etc. So, 2^x means to multiply 2 by itself x times.

Relationship Between Exponents and Logarithms

The logarithm is the inverse of an exponential function.

That is if you know that

y = 10^x

then you can take the log of both sides to solve for x.

\log_{10} y = \log_{10} 10^x = x

Note that the logarithm function has to correspond to the number being exponentiated.

When we use the number e we use the natural log \ln

y = e^x

\ln y = \ln e^x = x

For any other number b, if y = b^x then \log_b y = x