Math: Exponents and logarithms

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For any real number x and any integer exponent m,

xm = x*x*x*...*x

(m times). So exponentiation is a form of repeated multiplication.

The rules

xm * xn = xm+n xm / xn = xm-n (xm)n = xm*n

where

x-m = 1 / xm

follow from this definition. These rules also hold for any real exponents m and n.

What does the following expression when simplified?

23*24

23*24 = 23+4 = 27 = 128 = 8 * 16 = 23*24

The inverse function of

x = by

is the logarithm function

y = lnb(x)

where b is the base.

Common bases are

base 2, typically used for analyzing the efficiency of computer algorithms,
base 10, used for pH (acid-base) and dB (sound) measurement, and
base e, the natural base, where e is a transcendental number (that is, a number defined only by an infinite series expansion) that starts out 2.81...

So, if

x = by

then

y = lnb(x).

This provides a way to perform multiplication (or division) by table lookup and addition (or subtraction). What is the simplified or reduced value of the following expression?

ln2(128)

ln2(128) = ln2(27) = 7

For example, suppose that you wish to compute the quantity z where

z = x*y.

Take the logarithm of both sides to get

ln(z) = ln(x*y) = ln(x)+ln(y).

Now take the exponential (the inverse function of the logarithm function, or antilog function) of both sides to get

eln(z) = e(ln(x)+ln(y))

so that

z = e(ln(x)+ln(y)).

So, to multiply x times y,

find the logarithm of x,
find the logarithm of y,
add both logarithms, and
take the antilog, or exponential.

Just as

multiplication provides an easy way to do repeated addition, and
division provides an easy way to do repeated subtraction,
logarithms and antilogs (exponentials) provide an easy way to do repeated multiplications and/or divisions.

Set up the solution to the problem, "If the population of the world (about 10 billion people) doubles about every 100 years, how long ago were there only 2 people on earth?" as an algebraic expression (but do not evaluate the expression to a solution).