OBJECTIVE To explore how logarithms are used to solve exponential equations.

Origami is the traditional Japanese art of paper folding. The Japanese word origami literally means "paper folding" (from oru, meaning "folding," and kami, meaning "paper"). The goal of this art is to create an illustration of an object by folding a sheet of paper. Most of us have learned how to make paper airplanes or simple paper boats. One of the most common origami constructions is the crane shown here. But origami masters can fold paper into creative forms, including animals, birds, flowers, buildings, and towers. In this exploration we investigate the simplest type of origami: just folding a sheet of paper over and over again.

I. How to Make a Paper Tower

Take an ordinary sheet of paper and fold it in half. Now fold it in half again. Keep folding the paper in half. How many times are you able to fold it?

1. 1. When you fold the sheet of paper once, the result is a stack with two layers of paper; folding it twice results in a four-layer stack, and so on. Complete the table for the number of layers after $x$ folds.

      Number of folds $x$	$0$	$1$	$2$	$3$	$4$	$5$
      Layers $f(x)$	$1$

   2. The number of layers grows exponentially. What is the growth factor? What is the initial value? Find an exponential function $f(x) = Ca^x$ that models the number of layers after $x$ folds. $$f(x) = \color{red}{\fbox{    }} . \color{red}{\fbox{    }}^x$$
2. Suppose the sheet of paper we started with is $1/1000$ of an inch thick.
   1. How thick is the stack of paper after $10$ folds?

      Number of layers: $\underline{2^{\color{red}{\fbox{  }}}}$

      Thickness in inches: _______

   2. How thick is the stack of paper after $20$ folds?

      Number of layers: $\underline{2^{\color{red}{\fbox{  }}}}$

      Thickness in inches: _______

      Thickness in feet: _______

   3. How thick is the stack of paper after $50$ folds?

      Number of layers: $\underline{2^{\color{red}{\fbox{  }}}}$

      Thickness in miles: _______

II. How High a Tower?

How many times do you need to fold the paper to make a tower as tall as you are? Or as tall as the Empire State Building? To answer these questions, let's start with a sheet of paper that is $1/1000$ of an inch thick.

1. How many sheets of paper are needed to form a stack of paper with the given height?

   1 inch: _______

   1 foot: _______

   1 mile: _______

2. How many times do you need to fold the paper to get a stack of paper as tall as you are? Answer this question in the following steps:

   Express your height in inches: $h$ = _______

   Express your height in sheets of paper: $N$ = _______

   Solve the equation $2^x = N$ for $x$. Explain why your answer is the number of folds you need to get a stack your height. Can you actually fold the paper that many times?
3. How many times do we need to fold the paper to get a stack with the following heights?
   1. The height of the Empire State Building ($1250$ feet)
   2. The height of Mount Everest ($29,029$ feet)
   3. The distance to the moon ($240,000$ miles)
   4. The distance to the sun ($93,000,000$ miles)