Simple Nature - Light and Matter

Approximations to Exponents and Logarithms

It is often useful to have certain approxi-
mations involving exponents and logarithms.
As a simple numerical example, suppose that
your bank balance grows by 1% for two years
in a row. Then the result of compound in-
terest is growth by a factor of 1.01^2 = 1.0201,
but the compounding effect is quite small, and
the result is essentially 2% growth. That is,
1.01^2 ≈ 1.02. This is a special case of the
more general approximation

(1 +)p≈1 +p,

which holds for small values ofand is used
in example 4 on p. 408 relating to relativity.
Proof: Any real exponentp can be approx-
imated to the desired precision asp = a/b,
whereaandbare integers. Let (1+)p= 1+x.
Then (1 +)a= (1 +x)b. Multiplying out both
sides gives 1 +a+...= 1 +bx+..., where
...indicates higher powers. Neglecting these
higher powers givesx≈(a/b)≈p.
We have considered an approximation that
can be found by restricting thebaseof an ex-
ponential to be close to 1. It is often of interest
as well to consider the case where theexponent
is restricted to be small. Consider the base-e
case. One way of definingeis that when we
use it as a base, the rate of growth of the func-
tionex, for smallx, equals 1. That is,

ex≈1 +x

for smallx. This can easily be generalized to

other bases, sinceax=eln(a

x)

=exlna, giving

ax≈1 +xlna.

Finally, sinceex≈1 +x, we also have

ln(1 +x)≈x.

Problems 1019